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

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 τ 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.