Spectral-Petrov-Galerkin Method for Parabolic Problems Based on Darcy’s Law-Preserving

Darcy’s law is the fundamental equation describing the flow of a fluid through a porous medium. Combined with the principle of mass conservation, it leads to the diffusion Equation (e.g., the groundwater flow equation). In this paper, the Legendre-Petrov-Galerkin method is developed for solving the parabolic problem with Dirichlet boundary conditions based on Darcy’s law-preserving. This problem is transformed into an equivalent first-order system by introducing a flux based on Darcy’s law. Our scheme is based on the Legendre Galerkin method, and the right hand side term is processed using the Legendre/Chebyshev-Gauss-Lobatto points. The time direction is approximated by the Crank-Nicolson method. The algebraic system with a sparse coefficient matrix is obtained by selecting the appropriate basis function. Error estimate of the semi-discrete scheme is given by using Gronwall’s inequality (integral form) and Darcy’s law. Numerical examples show that our scheme has the high-order spectral accuracy and it preserves Darcy’s law.