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A Novel Meshfree Strategy for a Viscous Wave Equation with Variable Coefficients

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A Novel Meshfree Strategy for a Viscous Wave Equation with Variable Coefficients BRIEF RESEARCH REPORT published: 02 July 2021 doi: 10.3389/fphy.2021.701512 A Novel Meshfree Strategy for a Viscous Wave Equation With Variable Coefficients Fuzhang Wang 1,2, Juan Zhang 2*, Imtiaz Ahmad 3, Aamir Farooq 4 and Hijaz Ahmad 5 1College of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou, China, 2School of Computer Science and Technology, Huaibei Normal University, Huaibei, China, 3Department of Mathematics, University of Swabi, Swabi, Pakistan, 4Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad, Pakistan, 5Section of Mathematics, International Telematic University Uninettuno, Roma, Italy A one-step new general mesh free scheme, which is based on radial basis functions, is presented for a viscous wave equation with variable coefficients. By constructing a simple extended radial basis function, it can be directly applied to wave propagation by using the strong form-based mesh free collocation method. There is no need to deal with the time- dependent variable particularly. Numerical results for a viscous wave equation with variable coefficients show that the proposed mesh free collocation method is simple with accurate solutions. Edited by: Keywords: radial basis functions, wave equations, numerical simulation, meshfree methods, variable Umberto Lucia, Politecnico di Torino, Italy Reviewed by: 1 INTRODUCTION Lihua Wang, Tongji University, China The wave propagation exists in an interesting class of problems, such as the micro-scale heat transfer, Aatish Anshuman, seismic data acquisition and processing, etc. Acoustic wave modeling is an essential part of the Indian Institute of Technology technique of acoustic imaging [1]. Such problems have been widely investigated because of their Bombay, India Hanen Jrad, realistic physical background. Related works can be found in [2] and references therein. National Engineering School of Sfax, Since the governing equation of the problem considered is complex, and the corresponding Tunisia physical domain or the boundary conditions are tangle some, the analytical solutions are almost – *Correspondence: inaccessible for most practical engineering problems [3 7]. The development of accurate and fi Juan Zhang ef cient numerical methods becomes an important research topic in computational physics. [email protected] The earliest numerical methods for wave equations include the finite difference method [8], the finite element method [9], the boundary element method [10] and the finite volume method [11]. It Specialty section: should be noted that most of the existed numerical methods are still based on the finite difference This article was submitted to method (FDM) [12, 13], which lead to two-step finite difference approximations. More specifically, Interdisciplinary Physics, the finite difference method is used to deal with the time variable, and the rest procedures are finished a section of the journal by the other numerical methods. Coupled with the FDM, the radial basis function (RBF) collocation Frontiers in Physics methods are widely-used to many problems thanks to their mesh free features. Akers et al. [14] Received: 28 April 2021 investigated a radial basis function-finite differencing (RBF-FD) scheme which is applied to the Accepted: 08 June 2021 initial value problem of the Benjamin-Ono equation. Bhardwaj and Kumar [15] used the FDM-based Published: 02 July 2021 mesh less approach for the numerical solution of the nonlinear diffusion-wave equation. Based on Citation: the integrated radial basis functions and a finite difference scheme, the regularized long-wave Wang F, Zhang J, Ahmad I, Farooq A equations are investigated in [16, 17]. Oruc [18] examined numerical solutions of Zakharov- and Ahmad H (2021) A Novel Meshfree fi Strategy for a Viscous Wave Equation Rubenchik system by using radial basis function nite difference (RBF-FD) mesh less method and an With Variable Coefficients. explicit Runge-Kutta method. The time discretization is accomplished by means of an implicit Front. Phys. 9:701512. method based on the theta-weighted and finite difference methods, while the spatial discretization is doi: 10.3389/fphy.2021.701512 described with the help of the finite difference scheme derived from the local radial basis function Frontiers in Physics | www.frontiersin.org 1 July 2021 | Volume 9 | Article 701512 Wang et al. Novel Meshless Method method. Recently, Lu et al [19] used the radial basis function will lead to elliptic-type problems. Then the rest work is done by mesh less method to solve the irregular region interface problem. the other numerical methods. In order to improve the perplexing Ranocha et al [20] created new classes of fully-discrete two-step methods, we propose a direct one-step collocation conservative methods for several nonlinear dispersive wave method by using an extended RBFs in the following section. equations. In order to simplify the conventional two-step solution process, Netuzhylov and Zilian [21] proposed a space-time 3 FORMULATION OF THE EXTENDED mesh free collocation method for solving partial differential RADIAL BASIS FUNCTIONS equations by a consistent discretization in both space and time. Motivated by this work, we aim to provide a truly mesh The radial basis functions (RBFs) are extremely powerful tool for free method with one-step approximation, which is based on the solving partial differential equations. It is “isotropic” for RBFs, for a viscous wave equation with variable coefficients. Euclidean spaces. The Kansa’s method [22, 23] is one of the Compared with the traditional FDM-based two-step methods, most famous RBFs. It is also called the multi-quadrics RBF with the newly-proposed one-step direct mesh free method is truly the expression [24]. mesh free with easy implementation. The computational efficiency is excellent and the computational accuracy is 2 ϕ r 1 + εr (5) mainly determined by numerical methods for wave field mq j j simulation. ε The paper is organized as follows. In Section 2, formulation of where rj is the Euclidean distance between two points. is the RBF the direct radial basis function with space-time Euclidean shape parameter which can balance the solution accuracy. distance is briefly introduced. This is realized by considering As is known to all, a linear combination of RBFs can provide time-dependent variable as normal time-independent variable. the numerical solutions for boundary value problems governed Thus, the time-dependent variable can be treated easily during by elliptic partial differential equations. In literatures, this can be the whole solution process. Section 3 presents the methodology not directly used to solve boundary value problems governed by for a viscous wave equation with variable coefficients under initial parabolic or hyperbolic partial differential equations. condition and boundary conditions. Numerical results with To make the direct applications possible, the time variable is fi detailed discussions are presented for the accuracy of the treated equally with the space variables. More speci cally, we proposed mesh free method in Section 4. Section 5 concludes construct a simple direct radial basis function by combining the ( , ) this paper with some additional remarks. two-dimensional point x y and the one-dimensional time variable t as a three-dimensional point (x, y, t) for the space- time domain Ω × t. For collocation points generation, the time 2 PROBLEM DESCRIPTION variable is treated equally to the space variables x and y. For a simple example, the collocation points in the x− and y−directions fi / Let’s consider a viscous wave equation on a bounded two- are evenly chosen with neness hx hy 1 n. The time-step for − Δ / dimensional domain Ω ⊂ R2 the rest t direction is chosen as t 1 n. Similar to the traditional multi-quadric RBF, the extended z2u zu z radial basis function (ERBF) has the form Lu ≡ + α − ∇ · (Q∇u) − ∇ · (A∇u) f x, y, t, t > 0 zt2 zt zt 2 2 2 φ + ε2 2 + ε2 − + ε2 − + ε2 − (1) mq rj 1 rj 1 x xj y yj t tj where α is a nonnegative coefficient, f is a source term, Q (6) diag(q , q ) and A diag(a , a ) are diagonal, nonnegative 1 2 1 2 − diffusion tensors with q , q , a , a constant or variable where rj P Pj represents the distance between two space- 1 2 1 2 ( , , ) ( , , ) coefficients. Solution to Eq. 1 requires the extra initial conditions time coordinates P x y t and Pj xj yj tj . In literatures [25–27], the other researchers have proposed , , , , , ∈ Ω u x y 0 f1 x y x y (2) some space-time radial basis functions with different types. We utx, y, 0 f2x, y, x, y ∈ Ω (3) have done many investigations and found that they are not suitable to deal with the wave equations. and boundary conditions Bux, y, t f3x, y, t, x, y ∈ zΩ, t > 0 (4) 4 IMPLEMENTATION OF THE EXTENDED with f1(x, y), f2(x, y) and f3(x, y, t) the given functions. B is a boundary operator. One should seek for the numerical solution of RADIAL BASIS FUNCTION unknown function u(x, y, t). A variety of numerical methods have been investigated for 4.1 The Extended Radial Basis Function Eqs. 1–4. Most of them are based on the finite difference method, Solution Procedure which lead to two-step methods. More specifically, the finite To solve the problem described by Eqs. 1–4, we use the ERBF to difference method is used to deal with the time domain, which present the detailed procedure. Based on the definition of ERBF, Frontiers in Physics | www.frontiersin.org 2 July 2021 | Volume 9 | Article 701512 Wang et al. Novel Meshless Method TABLE 1 | The comparison of the absolute errors at t 0.5 s. TABLE 2 | The comparison of the absolute errors at t 0.9 s. Coordinates Analytical solutions ERBF Error Coordinates Analytical solutions ERBF Error (0.1, 0.1) 0.4145 0.4146 7.84 × 10−5 (0.1, 0.1) 1.0177 1.0177 4.62 × 10−5 (0.3, 0.3)−0.1583 −0.1584 6.35 × 10−5 (0.3, 0.3)−0.3887 −0.3875 1.26 × 10−3 (0.5, 0.5)−0.5123 −0.5123 1.90 × 10−5 (0.5, 0.5)−1.2580 −1.2593 1.28 × 10−3 (0.7, 0.7)−0.1583 −0.1584 6.35 × 10−5 (0.7, 0.7)−0.3887 −0.3875 1.26 × 10−3 (0.9, 0.9) 0.4145 0.4146 7.84 × 10−5 (0.9, 0.9) 1.0177 1.0177 4.62 × 10−5 Eqs 1–4 can be simulated by a one-step numerical method.
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