Six numerical schemes for parabolic initial boundary value problems with a priori bounded solution Original Research Article

The paper presents six new numerical finite-difference schemes to solve nonlinear parabolic initial boundary value problems with constraints imposed a priori on the solution. The first three proposed schemes are based on a special first-order approximation of the “diffusion” and “transport” terms combined with an unconditionally stable Gauss–Seidel-type iterative technique. We present a theoretical analysis of the method as applied to the diffusion wave equation and to the generalized diffusion wave approach. The fourth scheme allows for an accuracy enhancement at the expense of the computational cost. The scheme employs an adaptive grid which guarantees the second-order approximation or nearly so. An efficiency of the proposed techniques is demonstrated by the numerical simulations of flood in the eastern areas of Bangkok. Next, the first-order techniques are generalized to the Richardʹs type model of unsaturated porous medium flows, representing the general case of a nonlinear parabolic equation endowed with the space dependent bilateral constraints. The fifth and the sixth schemes are designed for space independent and space dependent constraints, respectively. We present a theoretical analysis of the methods. Finally, we verify the proposed schemes by methodological applications and analyze the convergence rate.