Solving hyperbolic partial differential equations using a highly accurate multidomain bivariate spectral collocation method

• Published in: Wave motion 2019-05, Vol.88, p.57-72
• Main Authors: Samuel, F.M., Motsa, S.S.
• Format: Article
• Language: English
• Subjects: Algorithms; Bivariate analysis; Bivariate polynomial interpolation; Collocation methods; Combinatorics; Computational efficiency; Computing time; Continuity equation; Discretization; Error analysis; Error bounds; Finite difference method; Hyperbolic PDEs; Initial conditions; Interpolation; Linear systems; Multidomain; Partial differential equations; Polynomials; Spectra; Spectral collocation
• ISSN: 0165-2125; 1878-433X
• DOI: 10.1016/j.wavemoti.2019.01.014

In this article, the non-overlapping grids based multidomain bivariate spectral collocation method is applied to solve hyperbolic partial differential equations(PDEs) defined over large time domains. The article is among the very first works which consider the multidomain approach with respect to the time interval for hyperbolic PDEs. The proposed method is based on decomposing the time domain into smaller non-overlapping subintervals and solving the PDE independently on each of these subintervals. In this study, we aim at showing that the reduction in the size of the computational domain at each subinterval guarantees accurate results within a short computational time. In the solution process, the approximate solutions of the PDEs are approximated using bivariate Lagrange interpolating polynomials. The PDEs are discretized in both time and space variables using the spectral collocation, unlike previous studies where spectral collocation method has been applied on space variable only and finite difference based discretization in the time variable and vice versa. The resulting linear systems of algebraic equations are then solved independently at each subinterval with the continuity equation being employed to obtain initial conditions in subsequent subintervals. Finally, the approximate solutions of the PDEs are obtained by matching the solutions on different subintervals along common boundaries. The new error bound theorems and proofs for bivariate polynomial interpolation using Gauss–Lobatto nodes given explain the advantages of the proposed solution algorithm. The effectiveness and accuracy of the proposed method are demonstrated by presenting error analysis and the computational time for the solution of well known hyperbolic PDEs that have been reported in the literature. The method can be adopted and extended to solve problems in real life that are modeled by hyperbolic PDEs.