Improved Trivariate Spectral Collocation method Mathematics Article (Article Sample)

Improved trivariate spectral collocation method of solution for two-dimensional nonlinear initial-boundary value problems via domain decomposition
F.M. Samuel1, 3 and S.S. Motsa1, 2
1 University of KwaZulu-Natal, School of Mathematics, Statistics, and Computer Science, Private Bag X01, Scottsville, 3209, South Africa;
felexmutua@gmail.com
2 University of Swaziland, Department of Mathematics, Private Bag 4, Kwaluseni, Swaziland;
sandilemotsa@gmail.com
3 Taita Taveta University, Department of Mathematics, 635-80300, Voi, Kenya.;
Abstract. In this article, we propose an accurate and computationally efficient overlapping grid based multi-domain trivariate spectral collocation method for solving two-dimensional nonlinear initial-boundary value problems over large time intervals. In the current solution approach, the quasi-linearization method is used to simplify the nonlinear PDEs. The spatial domain is decomposed into a sequence of overlapping subintervals of equal length whereas the time domain is broken into equal non-overlapping subintervals. Trivariate Lagrange interpolating polynomials constructed using Chebyshev-Gauss-Lobatto (CGL) points is used to approximate solutions to the nonlinear PDEs. A purely spectral collocation-based discretization is employed on the two space variables and the time variable on each subinterval to yield a system of linear algebraic equations that is solved. The PDEs are solved simultaneously across all subintervals in space but independently on each subinterval in time with the continuity condition been applied to obtain initial conditions in subsequent time subintervals. The numerical scheme is tested on typical examples of two dimensional nonlinear parabolic PDEs reported in the literature as a single equation or system of equations. Numerical results confirm that the proposed solution approach is highly accurate and computationally efficient when applied to solve two-dimensional initial-boundary value problems defined on large time intervals and large spatial domains when compared with the standard method on a single domain. In addition, it is demonstrated that the overlapping grids technique preserve the stability of the numerical scheme when solving fluid mechanics problems for large Reynolds numbers. The new error bound theorems and proofs on trivariate polynomial interpolation that we present support findings from the numerical simulations.
Keywords: Trivariate Lagrange Interpolating Polynomials, Spectral Collocation, Two-dimensional Parabolic PDEs, Overlapping Grid, Non-overlapping Grids, CGL Points.
Introduction
Spectral collocation-based methods, since their existence, have gained popularity in the numerical approximation of the solution of partial differential equations owing to their superior accuracy when applied to solve problems with smooth solutions [1]. They are particularly desirable for approximating solutions of nonlinear PDEs defined on regular geometries and they require a few numbers of grid points to achieve results with stringent accuracy [2]. Despite the benefits of the spectral collocation methods, review of the literature indicates that previous application of purely spectral collocation methods has focused on the solutions of ordinary differential equations and or partial differential equations involving two independent variables [3]. There exists extensive literature in the studies by Zhao et al. [4] and Tadmor [5] where the spectral collocation methods have been applied successfully on these types of problems and highly accurate results achieved in a computationally efficient manner have been reported. In a few noticeable exceptions, for instance, in [6] where spectral collocation methods have been applied to obtain numerical solutions of two-dimensional time-dependent PDEs, such has been achieved through the application of spectral collocation discretization on the space variables and finite difference discretization on the time variable. It is well known that finite difference methods require many grid points to yield accurate results which can hardly match those obtained when spectral collocation methods are applied on PDEs defined on simple geometries particularly if the underlying solutions are smooth. As observed in [7], the accuracy of a single domain based spectral collocation method deteriorates when the method is applied to solve PDE defined over a large time interval even with a large number of grid points. As shown by Motsa et al. [8], the utility of the spectral collocation method can be improved by decomposing the large time domain into smaller non-overlapping subintervals and solving the differential equation independently at each of these subintervals in time. Further, as it will be shown later, decomposing the spatial domains into a sequence of overlapping subintervals improves accuracy when solving differential equations defined over large spatial domains and presents a stable numerical scheme when solving a differential equation for very large or small values of the parameters. Motivated by these facts, we propose an overlapping grid based multi-domain trivariate spectral collocation method solving nonlinear two-dimensional time-dependent PDEs defined on large rectangular domains over large time intervals.