Stability and Error Estimates of High Order BDF-LDG Discretizations for the Allen-Cahn Equation

摘要

We construct high order local discontinuous Galerkin (LDG) discretizations coupled with third and fourth order backward differentiation formulas (BDF) for the Allen-Cahn equation. The numerical discretizations capture the advantages of linearity and high order accuracy in both space and time. We analyze the stability and error estimates of the time third-order and fourth-order BDF-LDG discretizations for numerically solving Allen-Cahn equation respectively. Theoretical analysis shows the stability and the optimal error results of theses numerical discretizations, in the sense that the time step tau requires only a positive upper bound and is independent of the mesh size h. A series of numerical examples show the correctness of the theoretical analysis. Comparison with the first-order numerical discretization illustrates that the high order BDF-LDG discretizations show good performance in solving stiff problems.

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