Modeling Tide–Induced Groundwater Response in a Coastal Confined

applied sciences

Article Modeling Tide–Induced Groundwater Response in a Coastal Conﬁned Aquifer Using the Spacetime Collocation Approach

Cheng-Yu Ku 1,2 , Chih-Yu Liu 2,* , Yan Su 3, Luxi Yang 3,* and Wei-Po Huang 1,2 1 Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung City 20224, Taiwan; [email protected] (C.-Y.K.); [email protected] (W.-P.H.) 2 Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung City 20224, Taiwan 3 Department of Water Resource and Harbor Engineering, College of Civil Engineering, Fuzhou University, Fuzhou 350108, Fujian, China; [email protected] * Correspondence: [email protected] (C.-Y.L.); [email protected] (L.Y.)

Received: 16 December 2019; Accepted: 2 January 2020; Published: 7 January 2020

Abstract: This paper presents the modeling of tide–induced groundwater response using the spacetime collocation approach (SCA). The newly developed SCA begins with the consideration of Trefftz basis functions which are general solutions of the governing equation deriving from the separation of variables. The solution of the groundwater response in a coastal confined aquifer with an estuary boundary where the phase and amplitude of tide can vary with time and position is then approximated by the linear combination of Trefftz basis functions using the superposition theorem. The SCA is validated for several numerical examples with analytical solutions. The comparison of the results and accuracy for the SCA with the time–marching finite difference method is carried out. In addition, the SCA is adopted to examine the tidal and groundwater piezometer data at the Xing–Da port, Kaohsiung, Taiwan. The results demonstrate the SCA may obtain highly accurate results. Moreover, it shows the advantages of the SCA such that we only discretize by a set of points on the spacetime boundary without tedious mesh generation and thus significantly enhance the applicability.

Keywords: tide–induced; groundwater; spacetime collocation approach; treﬀtz-basis functions; conﬁned aquifer; estuary

1. Introduction In coastal areas, tidal dynamics cause periodic fluctuation of the groundwater response in coastal aquifers. The dynamic behavior of the groundwater flow associated with the fluctuation of the piezometric head plays a crucial role in coastal hydrology [1–3]. Finding out the interaction between the tidal dynamics as well as the groundwater level may be useful for investigating coastal hydrogeology in large scale which is of importance to solve various coastal hydrogeological problems, such as deterioration of the marine environment, seawater intrusion, and the stability of coastal engineering structures with rapidly development in coastal areas [4,5]. Accordingly, considerable research has been devoted to understanding the tide–induced groundwater level fluctuations in a coast aquifer [6–8]. Analytical studies have been found to be common for understanding tidal variation affecting groundwater level in coastal areas [9–11]. Sun (1997) solved an analytical solution of groundwater response to tidal–induced boundary conditions in an estuary [12]. Jiao and Tang (1999) studied the groundwater response that consisted of unconfined and confined aquifers with a semi-permeable layer [13]. Jiao and Tang (2001) further proposed an exact solution for the groundwater

Appl. Sci. 2020, 10, 439; doi:10.3390/app10020439 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 439 2 of 21 response in a leaky confined aquifer [14]. Jeng et al. (2002) discussed the tidal propagation in a coupled unconfined and confined coastal aquifer system [15]. However, due to the mathematical complexity, it is a great challenge for finding exact analytical solutions of governing equations for different boundary conditions. As a result, the developed analytical solutions are usually limited and are restricted to simple problems with specific boundary conditions. Numerical approaches, such as finite elements and finite differences, approximate equations in discrete steps, both in time and in space [16,17]. The advantage of these approaches is no limit to the complexity of various coastal hydrogeological problems. However, the accuracy cannot be as accurate as the analytical method due to numerical errors [18]. Despite the success of several numerical methods for finding the solution, the traditional approaches need mesh generation, and hence tedious time calculation, to give an approximation to the solution [19]. There is therefore still a need for numerical schemes in the development of new advanced approaches. Meshfree approaches appear as a rival alternative to mesh generation techniques. The conception of meshfree approaches is to solve a problem by using arbitrary or unstructured points that are collocated in the problem domain without using any meshes or elements [20]. The utilization of the mesh-free approach to acquire approximate solutions is advantageous for problem domain involving arbitrary geometry [21,22]. The collocation Trefftz method (CTM) is one of meshfree approaches for dealing with boundary value problem (BVP) where solutions can be described as a combination of general solutions which satisfy the governing equation [23–25]. The CTM may release the difficulties for finding exact solutions of governing equations and remains the characteristics of high accuracy due to the adoption of the general solutions [26–28]. The original CTM is limited to homogeneous and stationary BVPs. Lately, a spacetime meshfree approach for analyzing transient subsurface flow in unsaturated porous media has been proposed by Ku et al. (2018) [29]. In this study, we present a pioneering work for numerical solutions of tide–induced groundwater response in a coast aquifer using the spacetime meshfree method. This paper presents the numerical solutions of tide–induced groundwater response using the spacetime collocation approach (SCA). The newly developed SCA begins with the consideration of the Trefftz basis functions. The basis functions have to satisfy the governing equation of the tide–induced groundwater response in a coastal confined aquifer with an estuary tidal–induced boundary. The phase, as well as the amplitude of tide, can then vary with time and position. The separation of variables is utilized to derive the general solutions to be the basis functions. For the discretization of the domain, arbitrary collocation points are assigned on the spacetime domain boundaries. The SCA was validated for several numerical examples with analytical solutions. The comparison of the results and accuracy for the SCA with the time-marching method is carried out. In addition, the SCA is adopted to evaluate the tidal and groundwater piezometer data at the Xing–Da port, Kaohsiung, Taiwan.

2. Problem Statement The governing equation of one–dimensional tide–induced groundwater response in a coastal conﬁned aquifer is expressed as follows.

∂2h ∂h T Lh = S in t. (1) ∂x2 − ∂t <

In the preceding equations, t denotes the spacetime domain, h denotes the head, t denotes < time, S is the storage coeﬃcient of the aquifer, T is the transmissivity of the aquifer, L denotes the leakage coeﬃcient, where L = K/b, K and b denote the vertical permeability and thickness of the semi-permeable layer. The initial condition for solving Equation (1) is expressed as

h(x, t = 0) = h0(x, t = 0), (2) Appl. Sci. 2020, 10, 439 3 of 21

where h0(x, t = 0) denotes the initial total head. The boundary conditions of Equation (1) are given as

t h(x, t) = D(x, t) on ΓD, (3)

t hn(x, t) = N(x, t) on ΓN, (4) t t where n is the normal direction, ΓD is the spacetime Dirichlet boundary, ΓN denotes the spacetime Neumann boundary, D(x, t) denotes the Dirichlet boundary condition, and N(x, t) denotes the Neumann boundary condition.

3. The Spacetime Collocation Approach The spacetime collocation scheme begins with the consideration of transient Trefftz basis functions which are also the general solutions of the governing equation. Thus, it may be necessary to derive general solutions for the tide–induced groundwater flow problem in a confined aquifer. To establish the transient Trefftz functions for this problem, the separation of variables is adopted by considering the transient solutions of the tide–induced groundwater flow problem.

h(x, t) = X(x)K(t). (5)

For simplicity, the following equations are considered.

d2X dK X00 = , and K = . (6) dx2 0 dt Inserting Equation (6) into Equation (1) can obtain the following equation.

TX00 K LXK = SXK0 in t. (7) − < Dividing by X(x)K(t) in Equation (7), we may yield

X00 S K L = λ, and 0 + = λ, (8) X T K T where λ is a separation constant. To evaluate the eigenvalue of Equation (8), the p constant is introduced to ensure this constant to be positive or negative value. Hence, we may consider λ = p2, λ = 0, and λ = p2. The derivation of Treﬀtz functions for one–dimensional groundwater ﬂow is listed − in AppendixA. According to the superposition theorem, the transient solution of this problem is described as the linear combination of the general solutions. Consequently, we may yield the following series expansion.

q q L L x x L t L t h(x, t) = A e T + B e− T + A xe S + B e S 01 " 01 02 − 02# − w Tp2 L Tp2 L P ( − )t+px ( − )t px + A1pe S + B1pe S − p=1 , (9) " # w Tp2+L Tp2+L P ( )t ( )t + A2pe− S cos(px) + B2pe− S sin(px) p=1 where w denotes the basis function order, A01, B01, A02, B02, A1p, B1p, A2p and B2p are unknowns to be found. From Equation (9), the complete basis functions, T, are acquired as

( q q 2 2 2 2 ) L L L L Tp L Tp L Tp +L Tp +L T x T x t t ( − )t+px ( − )t px ( )t ( )t T = e , e− , xe− S , e− S , e S , e S − , e− S cos(px), e− S sin(px) . (10) Appl. Sci. 2020, 10, 439 4 of 21

The Neumann boundary is given as

q q q L q L h L x L x L t ∂ = T + − T + S + ∂x A01 T e B01 T e A02e− B02 " − # w Tp2 L Tp2 L P ( − )t+px ( − )t px + A1ppe s + B1p( p)e s − . (11) p=1 − " # w Tp2+L Tp2+L P ( )t ( )t + A2ppe− s sin(px) + B2ppe− s cos(px) p=1 −

To evaluate the coeﬃcients in Equations (9) and (11), the collocation method is utilized. We may assign collocation points on the Dirichlet and Neumann boundary using Equations (9) and (11). Then, the system of linear algebraic equations may be assembled as follows.

q q  L L Tp2 L Tp2+L   T x1 T x1 ( − )t1+px1 ( )tt   e e− e S e− S sin(px1)   q q     2 ··· 2  f1  L x L x Tp L Tp +L      T 2 T 2 ( − )t2+px2 ( )t2  A    e e− e S e− S sin(px2)  01   f   ···    2   . . . . .  B01   .   ......    .   . . . .  .   .   q q 2  .  =  ,  L L Tp L Tp2+L  .    (12)  xn +n xn +n ( − )t +px ( )    v +   T 1 2 T 1 2 S n1+n2 n1+n2 S tn1+n2    n1 n2   e e e e− sin(pxn1+n2 )      ···  A2p   .   . . . . .    .   . . . . .  B   .   . . . . .  2p    q q  g +  q L q L Tp2 L Tp2+L  2n1 n2  L T x2n1+n2 L T x2n1+n2 ( − )t + +px + ( )t +   e nx e− nx pe S 2n1 n2 2n1 n2 nx pe− S 2n1 n2 cos(px )nx  T − T ··· 2n1+n2 Finally, the equation can be written as follows.

Pα = B, (13) where P is a J Q matrix from the basis functions, α is a Q 1 vector, B is a J 1 vector, J is the boundary × × × point number, Q is the terms of the basis function, which can be deﬁned as Q = 4 + 4w, t , t , , t n +n 1 2 ··· 2 1 2 are time, x , x , , x n +n are spatial coordinates, A , A , , A p, B p are unknowns to be evaluated, 1 2 ··· 2 1 2 01 01 ··· 2 2 f , f , , g n +n are Dirichlet boundary data, n and n are the boundary point number in time 1 2 ··· 2 1 2 1 2 and space domain, respectively. The unknown coeﬃcients are evaluated for solving Equation (13). To evaluate the tide–induced groundwater response, the inner points must be collocated in the spacetime domain. The tide–induced groundwater response at inner points may then be calculated by Equation (13). The schematic diagram for the one–dimensional tide–induced groundwater problems is

Appl.depicted Sci. 2020 in, Figure10, 439 1. 5 of 21

FigureFigure 1. TheThe schematic schematic flowchart ﬂowchart diagram diagram of of this this study.

(a)

(b)

Figure 2. Conventional one-dimensional domain and the spacetime domain: (a) conventional one-dimensional transient problem; (b) boundary points of the problem in the spacetime domain.

4. Numerical Examples Five numerical implementations are conducted to investigate the accuracy of the proposed method. The objective of each numerical example is depicted in Table 1. In numerical examples 1 and 2, we conduct the sensitivity analysis and results comparison with the analytical solution. In the numerical example 3 and 4, we compare the accuracy of the proposed method with that of the time–marching finite difference method (FDM). Furthermore, we consider an application example in numerical example 5, where a time series of real tidal fluctuation data recorded at Xing–Da port by the Ministry of Economic Affairs Water Resources Department are used to study tide–induced groundwater flow problems.

Appl. Sci. 2020, 10, 439 5 of 21

In this article, we consider a novel spacetime collocation scheme to model the transient tide–induced groundwater problems. On the basis of the spacetime collocation scheme, time is considered to be an independent variable [30,31]. Figure2a indicates a one-dimensional transient tide–induced groundwater problem. From Figure2a, denotes the space domain, L denotes the space dimension < f of the problem, and t f denotes the total elapsed time. It is clear that this problem is one-dimensional in both time as well as space. As depicted in Figure2b, the one-dimensional space domain turns into a two-dimensional spacetime domain. Hence, both initial as well as boundary values are assigned on the spacetime boundaries. Since the inaccessible boundary values are at the ﬁnal time, the novel proposed scheme may transform the one-dimensional unsteady state tide–induced groundwater problems into inverse BVPs in two-dimensions.Figure 1. The schematic flowchart diagram of this study.

(a)

(b)

Figure 2. Conventional one-dimensional domain and the spacetime domain: (a) conventional Figure 2. Conventional one-dimensional domain and the spacetime domain: (a) conventional one-dimensional transient problem; (b) boundary points of the problem in the spacetime domain. one-dimensional transient problem; (b) boundary points of the problem in the spacetime domain. 4. Numerical Examples 4. Numerical Examples Five numerical implementations are conducted to investigate the accuracy of the proposed method. The objectiveFive numerical of each implementations numerical example are is conducted depicted in to Table investigate1. In numerical the accuracy examples of the 1 and proposed 2, we conductmethod. the The sensitivity objective analysisof each andnumerical results example comparison is depicted with the in analytical Table 1. solution. In numerical In the examples numerical 1 exampleand 2, we 3 and conduct 4, we comparethe sensitivity the accuracy analysis of theand proposed results comparison method with with that ofthe the analytical time–marching solution. finite In dithefference numerical method example (FDM). 3 and Furthermore, 4, we compare we consider the accuracy an application of the proposed example method in numerical with examplethat of the 5, wheretime–marching a time series finite of realdifference tidal fluctuation method (FDM). data recorded Furthermore, at Xing–Da we consider port by the an Ministry application of Economic example Ainff airsnumerical Water Resourcesexample 5, Department where a time are series used of to re studyal tidal tide–induced fluctuation groundwater data recorded flow at Xing–Da problems. port by the Ministry of Economic Affairs Water Resources Department are used to study tide–induced groundwater flow problems.Table 1. The objective of each numerical example.

Numerical Examples Objective N A sensitivity analysis Numerical example 1 N Comparison with the analytical solution Numerical example 2 N Comparison with the analytical solution Numerical example 3 N Results comparison with the time–marching FDM N Accuracy comparison with the time–marching FDM Numerical example 4 N Investigation of diﬀerent leakage coeﬃcient N Evaluation of the tidal and piezometer data at the Numerical example 5 Xing–Da port, Kaohsiung, Taiwan Appl. Sci. 2020, 10, 439 6 of 21

4.1. Numerical Example 1 Since our numerical method is established from the CTM, the correctness of the numerical solutions may depend on the terms of the Treﬀtz functions. Hence, a sensitivity study is conducted. The governing equation of one–dimensional tide–induced groundwater response in a conﬁned aquifer is described as Equation (1). The data of the boundary are assigned by adopting the following analytical solution. r ( L+T )t ( 3L )t 2L h(x, t) = e− S sin x e− S cos( x). (14) − T

The initial data is r 2L h(x, t = 0) = cos( x) + sin x. (15) − T Figure2a depicts the configuration of the one–dimensional tide–induced groundwater problem. 2 The space dimension of the problem (L f ) is 1000 m. The transmissivity (T) is 1800 m /hr. The storage coefficient (S) is 2 10 4. The leakage coefficient (L) is 0.12/d. The elapsed time (t ) is 120 × − f hr. The numbers of the boundary and source points in this example are considered to be 150 and one. The initial data are provided on the bottom spacetime region. As for the boundary data, it is assigned on both right and left of the spacetime region, as illustrated in Figure2. To evaluate the accuracy of the proposed approach, the maximum absolute error (MAE) is evaluated by the following Appl. Sci. 2020, 10, 439 7 of 21 equation [32–34]. MAE = max he hn , (16) illustration is shown in Figure 2a. The initial and boundary| − | values are provided on the lateral and bottomwhere h eofdenotes the spacetime the exact region, solution as and displayedhn denotes in Figure the numerical 2b. There solution. exist one Figure source3 demonstrates point and the153 boundaryterms of the points. basis functionWe consider versus the the terms MAE. of Itthe seems basis that function the computed to be 10. resultsTo yield with the high field accuracy solution, may we considerbe obtained 2601 when inner the points terms ofdistributed the basis functionin the spacetime is greater thandomain. 10. To Figure further 4 investigatedepicts the the computed accuracy resultsof the SCA, with wethe considerexact solution. another It problem.seems that The the governing computed equation groundwater can be head described fluctuation as Equation using the (1). SCAThe initialmay closely condition agree is describedwith the analytical as solution. Figure 5 shows the correctness for the SCA. The -16 MAE associated with the SCA is of the orderh(x, oft = 8.90)× =10x.. It is obvious that the SCA may acquire (17) highly accurate results.

Figure 3. TheThe basis basis function function order order versus versus the maximum absolute error (MAE).

Figure 4. Result comparison of example 1. Appl. Sci. 2020, 10, 439 7 of 21 illustration is shown in Figure 2a. The initial and boundary values are provided on the lateral and bottom of the spacetime region, as displayed in Figure 2b. There exist one source point and 153 boundary points. We consider the terms of the basis function to be 10. To yield the field solution, we consider 2601 inner points distributed in the spacetime domain. Figure 4 depicts the computed results with the exact solution. It seems that the computed groundwater head fluctuation using the SCA may closely agree with the analytical solution. Figure 5 shows the correctness for the SCA. The MAE associated with the SCA is of the order of 8.9×10-16 . It is obvious that the SCA may acquire Appl. Sci. 2020, 10, 439 7 of 21 highly accurate results.

The right, as well as left boundary conditions are described as

L t h(x = L f , t) = xe− S , (18)

h(x = 0, t) = 0. (19)

The analytical solution is given as

L t h(x, t) = xe− S . (20)

The space dimension of the problem is 5000 m. The transmissivity is 1.25 m2/hr. The storage coefficient is 2 10 4. The leakage coefficient is 5 10 3/d. The elapsed time is 5 hr. The schematic × − × − illustration is shown in Figure2a. The initial and boundary values are provided on the lateral and bottom of the spacetime region, as displayed in Figure2b. There exist one source point and 153 boundary points. We consider the terms of the basis function to be 10. To yield the field solution, we consider 2601 inner points distributed in the spacetime domain. Figure4 depicts the computed results with the exact solution. It seems that the computed groundwater head fluctuation using the SCA may closely agree with the analytical solution. Figure5 shows the correctness for the SCA. The MAE associated with the SCA is of the order of 8.9 10 16. It is obvious that the SCA may acquire × − highly accurateFigure results. 3. The basis function order versus the maximum absolute error (MAE).

Figure 4. ResultResult comparison of example 1. Appl. Sci. 2020, 10, 439 8 of 21

Appl. Sci. 2020, 10, 439 8 of 21

FigureFigure 5. Absolute 5. Absolute error error of theof the results results computed computed with with the the analytical analytical solution solution at the at the final final time. time. 4.2. Numerical Example 2 4.2. Numerical Example 2 The governing equation of the numerical example 2 is given as Equation (1). The left boundary The governing equation of the numerical example 2 is given as Equation (1). The left boundary data are described as Equation (19). We consider the right boundary to be data are described as Equation (19). We consider the right boundary to be L+T ∂h(x =∂ L ,=t) +− h(xf L f ,t) ( L T()t )t = e−= eS Scoscosx.x . (21)(21) ∂n ∂n TheThe initial initial condition condition is is ( = )= = h x, t h(x0,t 0sin) xsin.x . (22)(22) TheThe exact exact solution solution is found is found as as ( L+T )t h(x t) = e S x , − L+sinT . (23) −( )t = S (23) Figure2a depicts the configuration of theh( one–dimensionalx,t) e sinx tide–induced. groundwater problem. The dimensionFigure of2a thedepicts space domain,the configuration the transmissivity, of the theone–dimensional storage coefficient, tide–induced the leakage coegroundwaterfficient, and the elapsed time are considered to be 5000 m, 1.25 m2/hr, 2 10 4, 5 10 3/d and 5 hr, respectively. problem. The dimension of the space domain, the transmissivity,× − the× storage− coefficient, the leakage Thecoefficient, boundary and initialthe elapsed data are time given are onconsidered the lateral to and be 5000 bottom m, of1.25 the m spacetime2/hr, 2×10 domain,-4 , 5×10 as-3 shown/d and 5 in Figurehr, respectively.2b. We place The a sourceboundary point and and initial 153 boundary data are given points on uniformly the lateral distributed and bottom on theof the boundary. spacetime Thedomain, Dirichlet as and shown Neumann in Figure boundary 2b. dataWe place are assigned a source on boundarypoint and points. 153 boundary We consider points the orderuniformly of thedistributed basis function on tothe be boundary. 10. To yield The the Dirichlet field solution, and Ne weumann collocate boundary 2601 inner data points. are assigned Figure6 on indicates boundary thepoints. computed We consider and the the exact order solutions. of the basis From function Figure 6to, thebe 10. computed To yield tide–inducedthe field solution, groundwater we collocate fluctuations2601 inner may points. be completely Figure 6 consistentindicates the with computed the exact solution.and the Figureexact solutions.7 shows the From correctness Figure for6, the 16 thecomputed SCA. Using tide–induced the SCA, the groundwater MAE is of the fluctuations order of 10− may. It is be obvious completely that the consistent SCA can acquirewith the high exact accuracysolution. performance. Figure 7 shows the correctness for the SCA. Using the SCA, the MAE is of the order of 10-16 . It is obvious that the SCA can acquire high accuracy performance. Appl. Sci. 2020, 10, 439 9 of 21 Appl. Sci. 2020, 10, 439 9 of 21 Appl. Sci. 2020, 10, 439 9 of 21

Figure 6. Result comparison of example 2. FigureFigure 6. Result 6. Result comparison comparison of example of example 2. 2.

Figure 7. Absolute error at final time. FigureFigure 7. Absolute 7. Absolute error error at final at final time. time. 4.3. Numerical Example 3 4.3.4.3. Numerical Numerical Example Example 3 3 The governing equation of numerical example 3 is represented by Equation (1). For this example, TheThe governing governing equation equation of numericalof numerical example example 3 is3 representedis represented by byEqua Equationtion (1). (1).For For this this two cases with different initial hydraulic heads are considered. Figure8 illustrates the schematic of example,example, two two cases cases with with different different initial initial hydrauli hydraulic headsc heads are areconsidered. considered. Figure Figure 8 illustrates 8 illustrates the the finite confined coastal aquifers. In Case A, the initial groundwater level is considered to be constant, schematicschematic of finiteof finite confined confined coastal coastal aquifers. aquifers. In CaseIn Case A, theA, theinitial initial groundwater groundwater level level is considered is considered hs = constant, as indicated= in Figure8a. In Case B, the initial head is considered to be hs = ax + h , to be constant, hs constant= , as indicated in Figure 8a. In Case B, the initial head is consideredmsl to to be constant, hs constant , as indicated in Figure 8a. In Case B, the initial head is considered to where=hs denotes+ the head of the seaside, a denotes the gradient of the hydraulic head in the seaside, be behs hax= axhmsl+ h, where, where hs h denotes denotes the thehead head of the of theseaside, seaside, a denotes a denotes the thegradient gradient of the of thehydraulic hydraulic and hmslsdenotesmsl the averageds mean sea level. As shown in Figure8b, the groundwater head may head in the seaside, and hmsl denotes the averaged mean sea level. As shown in Figure 8b, the linearlyhead increasein the seaside, as the distance and hmsl from denotes coastline the increases. averaged mean sea level. As shown in Figure 8b, the groundwatergroundwater head head may may linearly linearly increase increase as the as thedistance distance from from coastline coastline increases. increases. Appl.Appl. Sci.Sci.2020 2020, ,10 10,, 439 439 10 ofof 2121

(a) (b)

FigureFigure 8. 8.Schematic Schematic ofof ﬁnitefinitecoastal coastal conﬁnedconfined aquifer:aquifer: ((aa)) casecase A:A: constantconstant initialinitial hydraulichydraulic head;head; ((bb)) casecase B: B: inclined inclined initial initial hydraulic hydraulic head. head.

4.3.1.4.3.1. CaseCase A:A: ConstantConstant InitialInitial HydraulicHydraulic HeadHead InIn Case Case A, A, the the initial initial groundwater groundwater level level is is considered considered to to be be constant constant as as follows. follows. = = h(xh, t(x=,t 0)0 =) 0.0 . (24) The left tidal boundary data are The left tidal boundary data are 2π h(x = 0,t) = Asin( t) . (25) 2πt h(x = 0, t) = A sin( 0t). (25) t The boundary data on the inland side is described as 0 The boundary data on the inland side is describedh(x = ∞,t) as= 0 . (26)

Equation (26) describes that the tide hhas(x = nearly, t) no = 0.effect if x approaches far inland. The length (26) ∞ is 3000 m. The transmissivity is 31.25 m2/hr. The storage coefficient is 4.5×10-4 . The leakage coefficientEquation is (26)5×10 describes-3 /d. The that elapsed the tide time has is nearly 65 hr.no The eff ectamplitude if x approaches of the tidal farinland. change The (A) lengthis 2.5 m. is 3000 m. The transmissivity is 31.25 m2/hr. The storage coefficient is 4.5 10 4. The leakage coefficient The tidal period (t0) is 24.7 hr. The schematic illustration for case A is ×shown− in Figure 8a. is 5 10 3/d. The elapsed time is 65 hr. The amplitude of the tidal change (A) is 2.5 m. The tidal period × Chen− et al. (2013) applied the FDM to solve this problem [35]. To apply the FDM, discretization (tin0) isthe 24.7 spatial hr. The and schematic temporal illustration domains forwere case separately A is shown considered. in Figure8 a.For spatial discretization, the governingChen et equation al. (2013) of applied one-dimensional the FDM to solvetide–induc this problemed groundwater [35]. To apply response the FDM, for discretizationeach grid point in thewas spatial approximated and temporal using domains backward were difference separately formulas. considered. For Fortime spatial discretization, discretization, the implicit the governing scheme equationwas adopted of one-dimensional [35]. Since the tide–induced proposed method groundwater is one responseof the meshfree for each gridapproaches, point was this approximated problem is usingsolved backward by using dithefference boundary formulas. points For placed time on discretization, the spacetime the boundary. implicit scheme Therefore, was we adopted assign [ 35one]. Sincesource the point proposed and 163 method boundary is one points of the uniformly meshfree distributed approaches, on this the problem boundary. is solved The boundary by using data the boundaryare provided points on boundary placed on points. the spacetime We consider boundary. the or Therefore,der of the basis we assign function one to source be 25. point To obtain and 163 the boundaryfield solution, points we uniformly collocate distributed 2046 inner on points. the boundary. The numerical The boundary result of data our aremethod provided is then on boundarycompared points.with that We of consider the FDM. the Figure order of9 shows the basis the functionnumerica tol results be 25. Toadopting obtain thethe fieldFDM solution, and our wemethod. collocate It is 2046found inner that points. the tide–induced The numerical groundwater result of our fluctu methodations is then computed compared using with thatour ofmethod the FDM. agreed Figure well9 showswith those the numerical of the FDM. results adopting the FDM and our method. It is found that the tide–induced groundwater fluctuations computed using our method agreed well with those of the FDM. Appl. Sci. 2020, 10, 439 11 of 21 Appl. Sci. 2020, 10, 439 11 of 21

Figure 9. Result comparison of example 3 (case A).

4.3.2. Case Case B: B: Inclined Inclined Initial Hydraulic Head In Case B, the head of the seasideseaside is a function of distance x.. The initial head is given as

H L h(x,t = 0) = HL x . h(x, t = 0) = x.(27) (27) L f L f The boundary conditions at both left and right sides are given as The boundary conditions at both left and right sides are given as 2π h(x = 0,t) = Asin( t) , (28) 2tπ h(x = 0, t) = A sin( 0 t), (28) t = = 0 h(x L f ,t) HL . (29) h(x = L f , t) = HL. (29) = = −3 In this example, H 3 m , and HL / L 10 . Other parameters including the dimension of L f 3 In this example, HL = 3 m, and HL/L f = 10− . Other parameters including the dimension of the the space domain, the transmissivity, the storage coefficient, the leakage coefficient, the elapsed space domain, the transmissivity, the storage coefficient, the leakage coefficient, the elapsed time, the time, the amplitude of the tidal change, and the tidal period are the same as Case A. The schematic amplitude of the tidal change, and the tidal period are the same as Case A. The schematic illustration illustration for Case A is shown in Figure 8b. for Case A is shown in Figure8b. This example is solved using both FDM and our method. For the FDM analysis, the temporal This example is solved using both FDM and our method. For the FDM analysis, the temporal and spatial discretizations for this problem are separately considered. The implicit scheme and the and spatial discretizations for this problem are separately considered. The implicit scheme and the central difference approximation are considered for the temporal and spatial discretization, central difference approximation are considered for the temporal and spatial discretization, respectively. respectively. For the SCA analysis, we consider a source point and 163 boundary points. We For the SCA analysis, we consider a source point and 163 boundary points. We consider the basis consider the basis function order to be 25. Figure 10 depicts the results computed utilizing the FDM function order to be 25. Figure 10 depicts the results computed utilizing the FDM and our method. and our method. It is found that the tide–induced groundwater response computed using the It is found that the tide–induced groundwater response computed using the proposed method agreed proposed method agreed well with those of the FDM. well with those of the FDM. Appl. Sci. 2020, 10, 439 12 of 21 Appl. Sci. 2020, 10, 439 12 of 21

FigureFigure 10. 10. ResultResult comparison comparison of of example example 3 3 (case (case B). B).

4.4.4.4. Numerical Numerical Example Example 4 4 TheThe governinggoverning equationequation of of the the numerical numerical example exampl 4 ise given4 is asgiven Equation as Equation (1). The initial(1). The condition initial conditionis given as is Equation given as (24).Equation The left(24). tidal The boundary left tidal boundary head is given head as is given as π 22π h(x = 0,t) = A sin( t) . (30) h(x = 0, t) = A sin( t t). (30) t00 The boundary head on the inland side is described as The boundary head on the inland side is described as ∂ = h(x L f ,t) = 0 . (31) ∂h(x =∂Lnf , t) = 0. (31) The analytical solution [36] is found as ∂n The analytical solution [36] is found as −α 2π πS h(x,t) = Ae x sin( t − x) , α (32) t0 Tt0 αx 2π πS h(x, t) = Ae− sin( t x), (32) t 1 2Tt 1 2  2 0 2− α 0  1   L   2πS  L  α =  +    + .       1/2 (33) 2 T   Tt!021/2 T 1 L 2 2πS   L  α =  +  +  . (33)    Figure 8a depicts the schematic√ illustration2 T ofTt the0 one-dimensionalT  tide-induced groundwater problem. The dimension of the length is 3000 m. The transmissivity is 2000 m2/d. The storage Figure8a depicts the schematic illustration of the one-dimensional tide-induced groundwater -3 coefficientproblem. Theis 10 dimension . The amplitude of the length of the is 3000tidal m.change The is transmissivity 0.65 m. The istidal 2000 period m2/d. is The24 hr. storage The leakage coefficient3 is considered to be 0/d. The total elapsed time is considered to be 3 hr. coefficient is 10− . The amplitude of the tidal change is 0.65 m. The tidal period is 24 hr. The leakage coeffiForcient the is SCA considered analysis, to we be 0consider/d. The total 183 boundary elapsed time points is considered and one source to be 3 point. hr. We consider the basis Forfunction the SCA order analysis, for the weanalysis consider to be 183 15. boundary We also collocate points and 3721 one inner source points point. for We the consider validation. the Figurebasis function 11 indicates order the for thecomputed analysis and to beexact 15. Wesolutions also collocate [36]. From 3721 Figure inner points11, the for computed the validation. tide– inducedFigure 11 groundwater indicates the computedfluctuations and are exact consistent solutions with [36 the]. From exact Figure solution. 11, the Using computed the SCA, tide–induced the MAE -6 isgroundwater of the order fluctuationsof 10 . It is are obvious consistent that the with SCA the can exact acquire solution. high Using accuracy the performance. SCA, the MAE is of the 6 order of 10− . It is obvious that the SCA can acquire high accuracy performance. Appl. Sci. 2020, 10, 439 13 of 21 Appl. Sci. 2020, 10, 439 13 of 21

FigureFigure 11. 11. ComparisonComparison of of the the computed computed head head fluc ﬂuctuationtuation with with the the analytical analytical solution. solution.

ToTo further illustrate the correctness of the comp computeduted tide–induced head head fluctuations fluctuations from from the the proposedproposed SCASCA andand that that from from the the conventional conventional time–marching time–marching FDM, FDM, we conduct we conduct a comparison a comparison example examplein numerical in numerical example example 4. The governing 4. The governing equation equati is representedon is represented by Equation by (1).Equation The initial (1). The condition initial conditionis given as is Equation given as (24).Equation The left(24). tidal The boundaryleft tidal boundary head is given head as is given as π = = 2 h(x 0,t) Acos(2π t) . (34) h(x = 0, t) = A cos( t t). (34) t00 The boundary data on the inland side is described as The boundary data on the inland side is described as ∂ = h(x L f ,t) = 0 . (35) ∂h(x =∂Lnf , t) = 0. (35) The analytical solution is obtained as follows.∂n −α 2π πS The analytical solution is obtained as= follows.x − , h(x,t) Ae cos( t α x) (36) t0 Tt0 2π πS h(x t) = Ae αx ( t x) , − cos 1 2 ,1 2 (36)  2 t0 2− αTt0  1   L   2πS  L  α =   +    +  . (37)    1/2 2 T   Tt0!21/2 T 1 L 2 2πS   L  α =  +  +  . (37)    Figure 8a depicts the schematic√ illustration2 T ofTt the0 one–dimensT  ional tide–induced groundwater problem. The dimension of the length is 3000 m. The transmissivity is 2000 m2/d. The storage Figure8a depicts the schematic illustration of the one–dimensional tide–induced groundwater coefficient is 10-3 . The amplitude of the tidal change is 0.65 m. The tidal period is 24 h. The problem. The dimension of the length is 3000 m. The transmissivity is 2000 m2/d. The storage leakage coefficient3 is considered to be 0, 0.01, and 0.05/d. Therefore, the amplitude of groundwater coefficient is 10− . The amplitude of the tidal change is 0.65 m. The tidal period is 24 h. The leakage head is calculated for different leakage coefficient. The total elapsed time is considered to be 3 and 6 coefficient is considered to be 0, 0.01, and 0.05/d. Therefore, the amplitude of groundwater head is hr. calculated for different leakage coefficient. The total elapsed time is considered to be 3 and 6 hr. This example is solved utilizing both FDM and SCA. For the FDM analysis, the implicit scheme This example is solved utilizing both FDM and SCA. For the FDM analysis, the implicit scheme and the central difference approximation are considered for the temporal and spatial discretization, and the central difference approximation are considered for the temporal and spatial discretization, respectively. For the SCA analysis, we consider 183 boundary points and one source point. We respectively. For the SCA analysis, we consider 183 boundary points and one source point. We consider consider the basis function order for the analysis to be 15. To obtain the field solution, we collocate the basis function order for the analysis to be 15. To obtain the field solution, we collocate 3721 inner 3721 inner points. Figures 12 and 13 depict the computed results from the analytical solution [13], points. Figures 12 and 13 depict the computed results from the analytical solution [13], the Laplace the Laplace domain solution [37], the FDM, and our method at tf = 3 hr and tf = 6 hr , domain solution [37], the FDM, and our method at t f = 3 hr and t f = 6 hr, respectively. It is found respectively.that the computed It is found results that of tide–inducedthe computed groundwater results of tide–induced fluctuations groundwater using our method fluctuations closely using agree ourwith method the analytical closely solution. agree with Figure the analytical14 demonstrates solution. the Figure comparison 14 demonstrates of the accuracy the comparison with the FDM of andthe = accuracythe proposed with method the FDMt f =and3 hr thewith proposed the consideration method t off di3 ffhrerent with leakage the consideration coefficient. We, of therefore,different leakagecompare coefficient. the absolute We, error therefore, of our approachcompare andthe thoseabsolute of theerror FDM. of our From approach Figure 14 and, the those MAE of of the the FDM. From Figure 14, the MAE of the FDM is only of the order of 10-2 and 10-3 . However, the MAE of our approach may reach up to the order of 10-9 , as displayed in Figure 14. Table 2 shows Appl. Sci. 2020, 10, 439 14 of 21 Appl. Sci. 2020, 10, 439 14 of 21 the accuracy of our method with those of the conventional time-stepping approach using the FDM withthe accuracy the consideration of our method of different with those leakage of the coefficient. conventional From time-stepping Table 2, it approachis found usingthat the the MAEFDM remainswith the in consideration the order of of10 -5different to 10-9 leakage . It is obvious coefficient. that greatFrom agreementTable 2, it by is thefound proposed that the method MAE -5 -9 mayremains be achieved. in the order of 10 to 10 . It is obvious that great agreement by the proposed method may be achieved. Table 2. The MAE of the finite difference method (FDM) and the spacetime collocation approach (SCA).Table 2. The MAE of the finite difference method (FDM) and the spacetime collocation approach (SCA). The MAE Time (hr) The Leakage Coefficient (/d) The FDMThe MAE The SCA Time (hr) The Leakage Coefficient (/d) The FDM-2 The SCA-5 0.00 1.51×10 1.53 ×10 -2 -5 0.00 1.51×10 1.53 ×10 -3 -7 3 0.01 2.68 × 10 2.29×10 -3 -7 Appl. Sci. 2020, 10, 439 3 0.01 2.68 × 10 2.29×10 14 of 21 -3 -9 0.05 2.76 × 10 1.61×10 -3 -9 0.05 2.76 × 10 1.61×10 2 0.003 × -2 × -5 FDM is only of the order of 10− and 10− . However, the1.51 MAE10 of our 1.93 approach10 may reach up to the -2 -5 9 0.00 1.51×10 1.93 ×10 order of 10− , as displayed in Figure 14. Table2 shows the accuracy-3 of our method-6 with those of 6 0.01 7.91×10 1.56 × 10 the conventional time-stepping approach using the FDM with the-3 consideration-6 of different leakage 6 0.01 7.91×10 1.56 × 10 -3 5-9 9 coefficient. From Table2, it is found that0.05 the MAE remains7.39 in× the10 order 1.73 of ×1010− to 10− . It is obvious -3 -9 that great agreement by the proposed method0.05 may be achieved.7.39 × 10 1.73 ×10

Figure 12. Comparison of the groundwater head fluctuation computed with previous studies ( Figuretf = 3 hr 12.12.).Comparison Comparison of of the the groundwater groundwater head head ﬂuctuation fluctuation computed computed with previous with previous studies (t fstudies= 3 hr ).( tf = 3 hr ).

Figure 13. Comparison of the groundwater head ﬂuctuation computed with previous studies (t f = 6 hr).

Table 2. The MAE of the ﬁnite diﬀerence method (FDM) and the spacetime collocation approach (SCA).

The Leakage The MAE Time (hr) Coeﬃcient (/d) The FDM The SCA 0.00 1.51 10 2 1.53 10 5 × − × − 3 0.01 2.68 10 3 2.29 10 7 × − × − 0.05 2.76 10 3 1.61 10 9 × − × − 0.00 1.51 10 2 1.93 10 5 × − × − 6 0.01 7.91 10 3 1.56 10 6 × − × − 0.05 7.39 10 3 1.73 10 9 × − × − Appl. Sci. 2020,, 10,, 439 439 1515 of of 21

(a)

(b)

(c)

Figure 14. Comparison of the absolute error with the FDM and the SCA at t f = 3 hr:(a) L = 0.00/d; (b) L = 0.01/d; (c) L = 0.05/d. Appl. Sci. 2020, 10, 439 16 of 21

4.5. Numerical Example 5 Previous examples illustrate that our method is appropriately validated using the harmonic boundary conditions. Since the tidal variation is probably the major trigger inducing groundwater level fluctuation in coastal aquifers, the tidal fluctuation data are used to study tide–induced groundwater flow problems. This example is to evaluate the tidal and piezometer data at the Xing–Da port, Kaohsiung, Taiwan. The Xing–Da port is located in the Kaohsiung, southwestern Taiwan, as shown in Figure 15. The tidal data was from Yongan tidal station at 22◦49008.0000 N latitude and 120◦11051.0000 E longitude. The piezometer data was from Xing–Da groundwater observation well at 22◦51030.5200 N Appl.latitude Sci. 2020 and, 10 120, 439◦12 004.7700 E longitude [38]. 17 of 21

FigureFigure 15. 15. TheThe location location of of Xing–Da Xing–Da port port in in southwestern southwestern Taiwan. Taiwan.

The initial head is described as Equation (24). This study analyzed the tidal fluctuation data recorded at Xing–Da port from 00:00 p.m. on 22 December to 06:00 a.m. on 23 December 2018, by the Ministry of Economic Affairs Water Resources Department, as shown in Figure 16. It is found that the highest hourly tidal fluctuation reached 0.85 m, whereas the lowest hourly tidal fluctuation may reach 0.29 m. Hence, the left tidal boundary condition is described as − h(x = 0, t) = T(x), (38) where T(x) denotes the measured data by the Water Resources Agency, Ministry of Economic Affairs [38], as shown in Figure 16. The boundary head on the inland is described as

h(x = L f , t) = 0. (39)

According to the Engineering Geological Investigation Data Bank by the Central Geology Survey of the Ministry of Economic Affairs [38], the property of the confined aquifer and semi-permeable layers are sand and silt, respectively. Figure 17 shows the schematic of the numerical example 5. Figure 16. The tidal fluctuation and the measured groundwater response. The thickness of the confined aquifer and semi-permeable layer are 65 m and 15 m, as shown in Figure 17. The parameters used are listed in Table3. From Table3, the permeability of the confined aquifer and semi-permeable layer are 2 10 3 cm/s and 10 5 cm/s. The transmissivity is 4.68 m2/h. × − − The storage coefficient is 10 4. The leakage coefficient is 6 10 4/d. The dimension of the length is − × − 3000 m. The final elapsed time is 30 hr. The amplitude of the tidal change is 1 m. The tidal period is 24 h.

Figure 17. Schematic of numerical example 5. Appl.Appl. Sci.Sci. 20202020,, 1010,, 439439 1717 ofof 2121

Appl. Sci. 2020, 10, 439 17 of 21 FigureFigure 15.15. TheThe locationlocation ofof Xing–DaXing–Da portport inin southwesternsouthwestern Taiwan.Taiwan.

FigureFigure 16. 16. TheThe tidaltidal fluctuationfluctuation ﬂuctuation andand thethe meme measuredasuredasured groundwater groundwater response. response.

FigureFigure 17. 17. SchematicSchematic ofof numericalnumerical example example 5. 5. Table 3. Parameters used in the numerical example 5.

Types of Aquifer Parameter Value thickness 65 m confined aquifer permeability 2 10 3 cm/s × − (sand) transmissivity 4.68 m2/hr 4 storage coefficient 10− thickness 15 m semi-permeable layer permeability 10 5 cm/s (silt) − leakage coefficient 6 10 4 1/d × −

We consider 663 boundary points and a source point. The basis function order for the analysis is 15. To obtain the tide–induced groundwater ﬂuctuation, we collocate 18,631 inner points. Figure 18 demonstrates the comparison of measured data with the computed groundwater ﬂuctuation using our method at x = 0 m. Red line denotes the computed numerical results and the gray square denotes the measured data. It is found that the computed results of the tide–induced groundwater head using the proposed method may agree with the measured data. Although there may be a shift in time in Figure 18, the overall behavior of the tide–induced groundwater head may be captured using our proposed method. Appl. Sci. 2020, 10, 439 18 of 21 Appl. Sci. 2020, 10, 439 18 of 21

FigureFigure 18. ComparisonComparison of of the the computed computed groundwater groundwater response with measured data.

5. Discussion Discussion To investigateinvestigate the the proposed proposed method, method, five five numerical numerical examples examples are conducted. are conducted. Based Based on the on results, the results,the discussions the discussions are carried are outcarried as follows. out as follows. We conductconduct the the sensitivity sensitivity analysis analysis and and results results compared compared with thewith analytical the analytical solution solution in example in 16 example1 and example 1 and example 2. Results 2. obtainedResults obtained demonstrate demonstrate that the that MAE the is MAE in the is order in the of order10− .of It is10 obvious-16 . It is obviousthat the SCAthat the may SCA yield may highly yield accurate highly results.accurate To resu furtherlts. To illustrate further theillustrate advantages the advantages of the proposed of the proposedapproach, examplesapproach, 3 andexamples 4 are conducted3 and 4 toare investigate conducted the to accuracy investigate of temporal the accuracy discretization. of temporal From 5 9 discretization.the results of thisFrom study, the results we indicate of this that study, the MAEwe indicate of the SCAthat the remains MAE inof the the order SCA ofremains10− to in10 the− . We also compare the MAE of our method and those of the FDM. The MAE of the FDM is only on the order of 10-5 to 10-9 . We also compare the MAE of our method and those of the FDM. The MAE order of 10 2 and 10 3. It is significant that the proposed method may have more accurate numerical of the FDM− is only on− the order of 10-2 and 10-3 . It is significant that the proposed method may results than those of the conventional time–marching FDM. have more accurate numerical results than those of the conventional time–marching FDM. We propose the spacetime collocation approach for solving the tide–induced groundwater response. We propose the spacetime collocation approach for solving the tide–induced groundwater For the solving of the tide–induced groundwater response in a coastal confined aquifer with an estuary response. For the solving of the tide–induced groundwater response in a coastal confined aquifer tidal–loading boundary using the proposed method, the boundary points are placed in the spacetime with an estuary tidal–loading boundary using the proposed method, the boundary points are region. Hence, both initial as well as boundary values are considered as boundary conditions on placed in the spacetime region. Hence, both initial as well as boundary values are considered as the spacetime boundaries. The one–dimensional initial value problem can then be regarded as a boundary conditions on the spacetime boundaries. The one–dimensional initial value problem can two–dimensional inverse BVP. Accordingly, the transient problems for tide-induced head response in a then be regarded as a two–dimensional inverse BVP. Accordingly, the transient problems for coastal confined aquifer may be solved without utilizing the time-stepping techniques. Besides, more tide-induced head response in a coastal confined aquifer may be solved without utilizing the accurate numerical results for solving transient problems can be obtained. time-stepping techniques. Besides, more accurate numerical results for solving transient problems However, the SCA has to derive the general solutions as the basis functions for the governing can be obtained. equation of one–dimensional tide–induced groundwater response in a coastal confined aquifer utilizing However, the SCA has to derive the general solutions as the basis functions for the governing the separation of variables. The solutions of the tide–induced head response are then described as a equation of one–dimensional tide–induced groundwater response in a coastal confined aquifer series adopting the addition theorem. As a result, the proposed technique can only deal with the linear utilizing the separation of variables. The solutions of the tide–induced head response are then governing equation. described as a series adopting the addition theorem. As a result, the proposed technique can only Furthermore, another disadvantage is that the transformation of the original one–dimensional deal with the linear governing equation. problem into two–dimensional spacetime boundary value problem may increase the volume of Furthermore, another disadvantage is that the transformation of the original one–dimensional computing. Because the proposed technique is a boundary–type method, the points may be collocated problem into two–dimensional spacetime boundary value problem may increase the volume of only on the spacetime boundary of the domain. Since we do not need to place any collocation points computing. Because the proposed technique is a boundary–type method, the points may be inside the domain, the proposed method may be more efficient than other numerical approaches. collocated only on the spacetime boundary of the domain. Since we do not need to place any Although the array of the computation from two–dimensional spacetime boundary value problem may collocation points inside the domain, the proposed method may be more efficient than other be larger than the original one–dimensional problem, it is found that the benefits of the compromise numerical approaches. Although the array of the computation from two–dimensional spacetime for increasing the volume of the computation to obtain highly accurate results for tide–induced boundary value problem may be larger than the original one–dimensional problem, it is found that groundwater flow problem may be acceptable. the benefits of the compromise for increasing the volume of the computation to obtain highly accurate results for tide–induced groundwater flow problem may be acceptable. Appl. Sci. 2020, 10, 439 19 of 21

In addition, similar to other boundary-type meshless methods, the proposed method may also suﬀer from the numerical instability due to the ill-posed phenomenon of the meshless method for solving two- or three-dimensional problems.

6. Conclusions The spacetime collocation approach for solving the tide–induced groundwater response has been proposed. We carried out numerical problems to verify our method. The comparison of the results and accuracy for the spacetime collocation approach (SCA) with the time–marching finite difference method is carried out. The findings and contributions of this study are addressed. Furthermore, the SCA is adopted to evaluate the tidal and groundwater piezometer data at the Xing–Da port, Kaohsiung, Taiwan. The results from numerical examples depict that the SCA may obtain highly accurate results comparing to other numerical methods. Besides, the proposed method demonstrates the advantage of the SCA that we only discretize by a set of points on the spacetime boundary without tedious mesh generation. Hence, it may significantly enhance the applicability.

Author Contributions: C.-Y.K. provided the conceptualization and finalized the manuscript; C.-Y.L. wrote the paper and worked on the data curation; Y.S. analyzed the data; L.Y. worked on the mathematical development and numerical computations; W.-P.H. supervised the research. All authors have read and agreed to the published version of the manuscript. Funding: This study is partially supported by the Ministry of Science and Technology, Taiwan, the Republic of China (MOST 108-2621-M-019-008). Acknowledgments: The authors thank the Ministry of Science and Technology for the generous financial support. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A The derivation of Treﬀtz functions for one–dimensional groundwater ﬂow is expressed as the following description. Considering λ = p2, we may yield

Tp2 L px px ( − )t X(x) = c1e + c2e− , and K(t) = d1e S , (A1) where c1, c2 and d1 are constants. Inserting Equation (A1) into Equation (5) may acquire the following equation. Tp2 L Tp2 L ( − )t+px ( − )t px h(x, t) = A1e s + B1e s − (A2) where A1 and B1 are unknown coeﬃcients to be determined. 2 K0 Tp L 2 L Considering K = S− = 0, we may yield p = T . We may, therefore, obtain the following equation. K = d2, (A3) where d2 is constant. The following solution may then be obtained.

q q L L T x T x h(x) = A2e + B2e− , (A4) where A2 and B2 are unknown coeﬃcients to be determined. Considering λ = 0, we may yield the possible solutions. L t X(x) = c3x + c4, and K(t) = d3e− S , (A5) Appl. Sci. 2020, 10, 439 20 of 21

where c3, c4 and d3 are constants. Substituting Equation (A5) into Equation (5) may yield the following equation. L t L t h(x, t) = A3xe− S + B3e− S , (A6) where A and B are unknown coeﬃcients to be determined. Considering λ = p2, we yield the 3 3 − following solutions. Tp2+L ( )t X(x) = c5 cos px + c6 sin px, and K(t) = d4e− S , (A7) where c5, c6 and d4 are constants. Substituting Equation (A7) into Equation (5), we may obtain

Tp2+L Tp2+L ( )t ( )t h(x, t) = A4e− S cos(px) + B4e− S sin(px), (A8) where A4 and B4 are unknown coeﬃcients to be evaluated.

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