Stability Analysis of Quasi-variable Grids Cubic Spline Fourth-Order Compact Implicit Algorithms for Burger’s Type Parabolic PDEs

Abstract

In the present paper, we describe cubic spline combined with high accuracy compact formulation for computing approximate solution values to nonlinear convection-dominated diffusion equations, and apply the method to several variants of Burger’s type parabolic partial differential equations. A combination of cubic spline interpolating polynomial and quasi-variable grids sequence yields a two-level implicit compact computational scheme that falls in the range of increased accuracy. The grid stretching parameter plays an important role while considering the boundary layer phenomenon to convection-dominated parabolic equations. The discretization on the quasi-variable grid network yields a more accurate and precise solution than those obtained on a uniform grid network. It is easy to extend the proposed scheme to the singular equation with compact operators, preserving the order and accuracy. It is shown through stability analysis that the new method is stable with an accuracy of order four and two in the spatial and temporal direction, respectively. Numerical illustrations with generalized Burgers–Fisher equation, Burgers–Huxley equation, and some physically relevant parabolic partial differential equations corroborate the theoretical analysis.