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

Let Lh be the five-point finite difference operator which has O(h2) local truncation error at the points h and 1 − h next to the ends of interval [0,1] and O(h4) at the interior mesh points 2h,3h,…,1 − 2h. 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 Lh has been used in a number of publications to solve elliptic and parabolic equations. In the paper, Lh is applied to approximate the second derivatives uxx and vxx in the two diffusion equations put = auxx + ƒ(t,x,u,v) and qvt = bvxx + g(t,x,u,v). It is proved that the obtained semi-discrete scheme is O(h4) globally convergent. For approximation of the derivatives with respect to t, the O(τ2) 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.