A highly accurate trivariate spectral collocation method of solution for two-dimensional nonlinear initial-boundary value problems

Abstract In this paper, we propose a new numerical method namely, the trivariate spectral collocation method for solving two-dimensional nonlinear partial differential equations (PDEs) arising from unsteady processes. The problems considered are nonlinear PDEs defined on regular geometries. In the current solution approach, the quasi-linearization method is used to simplify the nonlinear PDEs. The solutions of the linearized PDEs are assumed to be trivariate Lagrange interpolating polynomials constructed using Chebyshev Gauss-Lobatto (CGL) points. A purely spectral collocation-based discretization is employed on the two space variables and the time variable to yield a system of linear algebraic equations that are solved by iteration. The numerical scheme is tested on four typical examples of nonlinear 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 small time intervals and hence it is a reliable alternative numerical method for solving this class of problems. The new error bound theorems and proofs on trivariate polynomial interpolation that we present support findings from the numerical simulations.

[1]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[2]  T. Sauer,et al.  On the history of multivariate polynomial interpolation , 2000 .

[3]  S. Abbasbandy,et al.  Numerical study of magnetohydrodynamics generalized Couette flow of Eyring-Powell fluid with heat transfer and slip condition , 2016 .

[4]  A. Borhanifar,et al.  Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method , 2010, Comput. Math. Appl..

[5]  Michael Revers.  On the approximation of certain functions by interpolating polynomials , 1998, Bulletin of the Australian Mathematical Society.

[6]  A. Refik Bahadir,et al.  A fully implicit finite-difference scheme for two-dimensional Burgers' equations , 2003, Appl. Math. Comput..

[7]  Mujeeb R. Malik,et al.  A spectral collocation method for the Navier-Stokes equations , 1985 .

[8]  H. E. Salzer,et al.  Converting interpolation series into Chebyshev series by recurrence formulas , 1976 .

[9]  T. A. Zang,et al.  Spectral Methods for Partial Differential Equations , 1984 .

[10]  Elyas Shivanian,et al.  Analysis of the spectral meshless radial point interpolation for solving fractional reaction-subdiffusion equation , 2018, J. Comput. Appl. Math..

[11]  Elyas Shivanian,et al.  The spectral meshless radial point interpolation method for solving an inverse source problem of the time-fractional diffusion equation , 2018, Applied Numerical Mathematics.

[12]  S. Abbasbandy,et al.  Analysis of steady flows in viscous fluid with heat/mass transfer and slip effects , 2012 .

[13]  Rongpei Zhang,et al.  The new numerical method for solving the system of two-dimensional Burgers' equations , 2011, Comput. Math. Appl..

[14]  Kenan Yildirim.  A Solution Method for Solving Systems of Nonlinear PDEs , 2012 .

[15]  R. Bellman,et al.  Quasilinearization and nonlinear boundary-value problems , 1966 .

[16]  Ali H. Bhrawy,et al.  A highly accurate collocation algorithm for 1 + 1 and 2 + 1 fractional percolation equations , 2016 .

[17]  Saeid Abbasbandy,et al.  Multiple solutions of mixed convection in a porous medium on semi-infinite interval using pseudo-spectral collocation method , 2011 .

[18]  I. Prigogine,et al.  Symmetry Breaking Instabilities in Dissipative Systems. II , 1968 .

[19]  Yujiang Wu,et al.  Chebyshev-Legendre pseudo-spectral method for the generalised Burgers-Fisher equation , 2012 .

[20]  A. Polyanin,et al.  Handbook of Nonlinear Partial Differential Equations , 2003 .

[21]  W. Ang,et al.  The two-dimensional reaction–diffusion Brusselator system: a dual-reciprocity boundary element solution , 2003 .

[22]  E. Tadmor.  A review of numerical methods for nonlinear partial differential equations , 2012 .

[23]  Ali J. Chamkha,et al.  HEAT AND MASS TRANSFER IN STAGNATION-POINT FLOW OF A POLAR FLUID TOWARDS A STRETCHING SURFACE IN POROUS MEDIA IN THE PRESENCE OF SORET, DUFOUR AND CHEMICAL REACTION EFFECTS , 2010 .

[24]  Marta García-Huidobro,et al.  Vector p-Laplacian like operators, pseudo-eigenvalues, and bifurcation , 2007 .

[25]  E. H. Twizell,et al.  A second-order scheme for the “Brusselator” reaction–diffusion system , 1999 .

[26]  Jafar Biazar,et al.  An approximation to the solution of the Brusselator system by Adomian decomposition method and comparing the results with Runge-Kutta method , 2007 .

[27]  F. Khani,et al.  New exact solutions of the Brusselator reaction diffusion model using the exp-function method. , 2009 .

[28]  Lijun Yi,et al.  Legendre spectral collocation method for second-order nonlinearordinary/partial differential equations , 2013 .

[29]  Elyas Shivanian,et al.  An improved spectral meshless radial point interpolation for a class of time-dependent fractional integral equations: 2D fractional evolution equation , 2017, J. Comput. Appl. Math..

[30]  Gregor Kosec,et al.  Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations , 2012 .

[31]  C. Streett,et al.  Improvements in spectral collocation discretization through a multiple domain technique , 1986 .

[32]  Charles B. Dunham,et al.  Rate of convergence of discretization in Chebyshev approximation , 1981 .

[33]  Pascalin Tiam Kapen,et al.  An accurate shock-capturing scheme based on rotated-hybrid Riemann solver , 2016 .