Multi-domain multivariate spectral collocation method for (2+1) dimensional nonlinear partial differential equations

• Published in: Partial differential equations in applied mathematics : a spin-off of Applied Mathematics Letters Vol. 6; p. 100440
• Main Authors: Mkhatshwa, M.P., Khumalo, M., Dlamini, P.G.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2022; Elsevier
• Subjects: (2+1) dimensional PDEs; Chebyshev–Gauss–Lobatto points; Kronecker tensor product; Multi-domain; Multivariate spectral quasilinearisation method; Trivariate Lagrange interpolation; Kronecker tensor product; (2+1) dimensional PDEs; Multi-domain; Trivariate Lagrange interpolation; Chebyshev–Gauss–Lobatto points; Multivariate spectral quasilinearisation method
• ISSN: 2666-8181; 2666-8181
• DOI: 10.1016/j.padiff.2022.100440

The novelty of this work rests upon the use of the domain partitioning technique in time variable when discretizing the domain of solution in spectral collocation algorithm. The single domain multivariate spectral collocation based methods have been proven to be effective in solving time-dependent partial differential equations (PDEs) defined over small time domains. However, there is a significant loss of accuracy as time computational domain proliferates and also when the number of grid nodes approaches a definite particular number. Therefore, the establishment of the new innovative multi-domain multivariate spectral quasilinearisation method (MDMV-SQLM) is described for the purpose of solving (2+1) dimensional nonlinear PDEs defined on large time intervals. The main output of this study is confirmation that minimizing the size of time computational domain at each subinterval assures sufficiently accurate results that are attained using minimal number of nodal points and less computational time. The solution algorithm involves partitioning the time domain into multiple non-overlapping sub-domains, simplification of the nonlinear PDEs using the quasilinearisation method and assumption of approximate solutions using triple Lagrange interpolating polynomials with Chebyshev–Gauss–Lobatto (CGL) points. The multi-domain spectral collocation procedure is executed on the linear systems of algebraic equations, where the subsequent matrix systems are solved separately in every time sub-interval with the continuity equation essentially used in obtaining initial conditions in the next subintervals. MATLAB software is used to implement the solution algorithm and numerical results are demonstrated graphically and in tabular form. To highlight the efficaciousness and accuracy of the MDMV-SQLM, error estimates, condition numbers and computational time are presented for well known (2+1) dimensional nonlinear initial-Dirichlet boundary value problems. The adoption of the domain decomposition technique is efficacious in suppressing the numerical challenges linked to large matrices and ill-conditioned nature of the resulting coefficient matrix. Also, the communicated results confirm that the numerical scheme is computationally cheap, fast and yield extremely accurate and stable results with the aid of fewer number of grid points for large time domains.