A new high accuracy off-step cubic spline approximations on a quasi-variable mesh for the system of nonlinear parabolic equations in one space dimension

We study a new two-level implicit method of order two in time and three in space based on two off-step points and an in-between point for the system of 1D nonlinear parabolic equations on a quasi-variable mesh. The proposed method is derived directly from the consistency condition of cubic spline polynomial approximation. The method is unconditionally stable, when tested on a model equation. We solve the Fisher-Kolmogorov equation, the Kuramoto-Sivashinsky equation, coupled Burgers' and the Burgers-Huxley equations to demonstrate the usefulness of the proposed method. The numerical results confirm the stability character of the method for large Reynolds number.