O( τ2 + h4) finite difference scheme for decoupled system of two quasilinear parabolic equations

Abstract

Let L h be the five-point finite difference operator which has O( h 2) local truncation error at the points h and 1 − h next to the ends of interval [0,1] and O( h 4) at the interior mesh points 2 h,3 h,…,1 − 2 h. The operator generates the pentadiagonal coherent matrix which is not of positive type and not diagonally dominant. However, the matrix satisfies the maximum principle. The operator L h has been used in a number of publications to solve elliptic and parabolic equations. In the paper, L h is applied to approximate the second derivatives u xx and v xx in the two diffusion equations pu t = au xx + ƒ(t,x,u,v) and qv t = bv xx + g( t, x, u, v). It is proved that the obtained semi-discrete scheme is O( h 4) globally convergent. For approximation of the derivatives with respect to t, the O( τ 2) trapezoidal rule is used. The fully discrete scheme obtained in this way constitutes a system of algebraic equations associated with a pentadiagonal matrix. To solve this system of equations an implicit iterative method based on an algorithm for pentadiagonal matrices is proposed. Numerical results illustrating the method are presented.