Lod generalized trapezoidal formula schemes for parabolic differential equations in two space dimensions

Abstract

We describe locally one-dimensional (LOD) time integration schemes for parabolic differential equations in two space dimensions, based on the generalized trapezoidal formulas (GTF(α)). We describe the schemes for the diffusion equation with Dirichlet and Neumann boundary conditions, for nonlinear reaction-diffusion equations, and for the convection-diffusion equation in two space dimensions. The obtained schemes are second order in time and unconditionally stable for all α ∈ [0, 1]. Numerical experiments are given to illustrate the obtained schemes and to compare their performance with the better known LOD Crank-Nicolson scheme. While the LOD Crank-Nicolson scheme can give unwanted oscillations in the computed solution, our present LOD-GTF(α) schemes provide both stable and accurate approximations for the true solution.

Keywords:

• Parabolic differential equations
• Two space dimensions
• LOD schemes
• Crank-Nicolson scheme
• Generalized trapezoidal formulas
• Unconditional stability
• Nonlinear reaction-diffusion
• Convection-diffusion

^*Corresponding author.

^*Corresponding author.

Notes

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