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Weighted Scheme and the Galerkin Method

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Weighted Scheme and the Galerkin Method mathematics Article On the Numerical Simulation of HPDEs Using q-Weighted Scheme and the Galerkin Method Haifa Bin Jebreen *,† and Fairouz Tchier † Department of Mathematics, College of Science, King Saud University, Riyadh 11 551, Saudi Arabia; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work. Abstract: Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the q-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis. Keywords: interpolating scaling functions; hyperbolic equation; Galerkin method 1. Introduction Partial differential equations (PDEs) are ubiquitous in mathematically scientific fields and play an important role in engineering and physics. They arise from many purely math- ematical considerations, such as the calculus of variations and differential geometry. One of Citation: Bin Jebreen, H.; Tchier, F. On the momentous subclasses of PDEs is the hyperbolic partial differential equations (HPDEs). the Numerical Simulation of HPDEs HPDEs are used to model many phenomena such as biology, industry, atomic physics, and Using q-Weighted Scheme and the aerospace [1–3]. Telegraph and wave equations are the most famous types of HPDEs that Galerkin Method. Mathematics 2021, 9, are enforceable in various fields such as random walk theory, wave propagation, and signal 78. https://doi.org/10.3390/ analysis [2,4]. math9010078 In this study, we construct and analyze a numerical algorithm based on the q-weighted Received: 23 November 2020 method, and the Galerkin method to solve the HPDEs Accepted: 28 December 2020 ( ) + ( ) + ( ) = ( ) 2 [ ] 2 [ ] Published: 31 December 2020 wt x, t a1wx x, t a2w x, t f x, t , x 0, 1 , t 0, T , (1) with boundary condition Publisher’s Note: MDPI stays neu- w( t) = g(t) t 2 [ T] tral with regard to jurisdictional clai- 0, , 0, , (2) ms in published maps and institutio- and initial condition nal affiliations. w(x, 0) = h(t), x 2 [0, 1]. (3) Furthermore, we assume that f , g, and h are the known functions, and also a1 and a2 are real constants. Copyright: © 2020 by the authors. Li- In this paper, we attempt to apply an efficient scheme that has not been used before censee MDPI, Basel, Switzerland. to solve such problems. The method includes three steps. In the first step, we use the This article is an open access article q-weighted method to break the time interval into a finite number of time steps. At each distributed under the terms and con- time step, we obtain a linear ordinary differential equation. In the second step, the obtained ditions of the Creative Commons At- ODE is solved using the Galerkin method. To do this, interpolating scaling functions are tribution (CC BY) license (https:// used. In comparison to the scaling function arising from multiresolution analysis (MRA), creativecommons.org/licenses/by/ interpolation scaling functions (ISFs) have some properties that make them attractive. 4.0/). Mathematics 2021, 9, 78. https://doi.org/10.3390/math9010078 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 78 2 of 13 These characteristics include the flexible zero moments, a compact support, orthonormality, and having a closed-form. The most important property of these bases is the interpolation. This property is useful to avoid integrals to find coefficients in expansions. At the last step, the linear algebraic system obtained from the second one must be solved using an appropriate technique. Stability, consistency, and convergence analysis are investigated, and numerical tests guarantee their validity. Numerous studies proposed a variety of numerical and analytical solutions to HPDEs. Doha et al. [5] proposed a numerical method based on the collocation method for solv- ing a system consisting of such equations. In [6], the spectral-Galerkin method is pro- posed to solve this equation. Singh et al. [7] solved one-dimensional HPDEs with the initial and boundary conditions (2) and (3), using an algorithm based on Chebyshev and Legendre multiwavelets. Dehghan et al. [8] introduced a numerical scheme based on the cubic B-spline scaling functions to solve (1) with nonlocal conservation conditions. Bin Jebreen et al. [4] proposed an efficient method based on the wavelet Galerkin method to solve the Telegraph equation, and the collocation method based on interpolating scaling functions is also used for solving this equation in [3]. For study, we refer the readers to [9,10]. The outline of the paper is as follows. Section2 is devoted to the brief introduction to the interpolating scaling function. Mixed q-weighted scheme and Galerkin method based on interpolating scaling functions are used to solve the desired equation and the stability, consistency, and convergence analysis are also investigated in Section3. Section4 is devoted to some numerical examples to show the ability and accuracy of the proposed method. 2. Interpolating Scaling Functions To derive a set of bases that covers the multiresolution analysis conditions, Alpert et al. [11] introduced a set of functions to generate the nested spaces r ¥ fVJ gJ=0 2 L2([0, 1]) using piecewise polynomial bases of degree less than r ≥ 0 (the + J multiplicity parameter). Let J 2 Z [ f0g be given. Putting BJ := f0, ... , 2 − 1g, and R := f0, 1, ··· , r − 1g, there is a sequence of nested subspaces that are spanned by r k k J + VJ := SpanffJ,b := f (2 x − k), b 2 BJ, k 2 Rg ⊂ L2([0, 1]), r ≥ 0, J 2 Z [ f0g, k by means of the Interpolating scaling functions ff gk2R, introduced by Alpert using the Legendre polynomials. Assume that Pr is the Legendre polynomial of order r and ftkgk2R are the roots of Pr. Let fwkgk2R be the Gauss–Legendre quadrature weights [3,11]. We determine ISFs as follows: ( q 2 k Lk(2x − 1), x 2 [0, 1], f (x) = wk 0, o.w, where fLk(x)gk2R are the Lagrange polynomials r−1 x − t L (x) = i . k ∏ − i=0,i6=k tk ti k k0 h i = 0 0 h i These bases fulfill the orthonormality relation fJ,b, fJ,b0 db,b dk,k where ., . de- notes the L2-inner product on W := [0, 1]. Assume that [b2B IJ,b is a uniform finite discretization of W. Here, the subintervals J r IJ,b := [xb, xb+1] are specified by points xb := b/(2 ). To project a function into VJ , we r 2 r introduce the orthogonal projection PJ that maps L (W) onto the subspace VJ . Utilizing this projection, every function p 2 L2(W) can be represented in the form r k k p ≈ PJ (p) = ∑ ∑ pJ,bfJ,b. (4) b2BJ k2R Mathematics 2021, 9, 78 3 of 13 k Due to the orthonormality of the bases, it is easy to prove that the coefficients pJ,b can be obtained by hp, fk i = R p(x)fk (x)dx. To avoid integration, we apply the J,b IJ,b J,b interpolation property of ISFs [11,12], via r w t + 1 pk ≈ 2−J/2 k p 2−J( k + b) , b 2 B , k 2 R. (5) J,b 2 2 J r r Given r-times continuously differentiable function p 2 C (W), the projection PJ (p) is bounded by means of L2-inner product as 2 kPr( ) − k ≤ −Jr j (r)( )j J p p 2 r sup p x . (6) 4 r! x2[0,1] r According to this relation, the projection PJ is convergent when J or r increases. To study more details, we refer the readers to [13]. Consequently, this projection is conver- gent with the rate of O(2−Jr). r T We determine the vector function FJ := [Fr,J,0, ··· , Fr,J,2J −1] with = [ 0 ··· r−1] Fr,J,b : fJ,b, , fJ,b includes the scaling functions and called multi-scaling functions. k The approximation (4) can be rewritten using the vector P whose entries are Pbr+k+1 := pJ,b as follows: r T r PJ (p) = P FJ, (7) where P is a vector of dimensional N := r2J. To approximate a higher-dimensional function, the building blocks of the bases can be r,2 r r 2 utilized. In this regard, one can consider the subspace VJ := VJ × VJ ⊂ L (W × W) that is spanned by k k0 0 0 ffJ,bfJ,b0 : b, b 2 BJ, k, k 2 Rg. In order to derive an approximation of two-dimensional function p 2 L2(W × W), we r apply the projection operator PJ , viz. r r T r p(x, t) ≈ PJ (p)(x, t) = FJ (x)PFJ(t), (8) where components of the square matrix P of order N are obtained by r r −J wk0 wk −J −J 0 P 0 0 ≈ 2 p 2 (tˆ + b), 2 (tˆ 0 + b ) , (9) rb+(k+1),rb +(k +1) 2 2 k k (2r) and tˆk = (tk + 1)/2. If C (W × W) 3 p : W × W ! R, we can show that the error can be bounded as follows: 21−rJ 21−Jr kPr p − pk ≤ M 2 + , (10) J max 4rr! 4rr! where Mmax is a constant [12] as follows: ( ) ¶r ¶r ¶2r M = j ( )j j ( )j j ( 0 0)j max max sup r p x, t , sup r p x, h , sup r r p x , h .
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