Entropy dissipative higher order accurate positivity preserving time-implicit discretizations for nonlinear degenerate parabolic equations
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We develop entropy dissipative higher order accurate local discontinuous Galerkin (LDG) discretizations coupled with Diagonally Implicit Runge-Kutta (DIRK) methods for nonlinear degenerate parabolic equations with a gradient flow structure. Using the simple alternating numerical flux, we construct DIRK-LDG discretizations that combine the advantages of higher order accuracy, entropy dissipation and proper long-time behavior. The implicit time-discrete methods greatly alleviate the time-step restrictions needed for the stability of the numerical discretizations. Also, the larger time step significantly improves computational efficiency. We theoretically prove the unconditional entropy dissipation of the implicit Euler-LDG discretization. Next, in order to ensure the positivity of the numerical solution, we use the Karush-Kuhn-Tucker (KKT) limiter, which couples the positivity inequality constraint with higher order accurate DIRK-LDG discretizations using Lagrange multipliers. In addition, mass conservation of the positivity-limited solution is ensured by imposing a mass conservation equality constraint to the KKT equations. The unique solvability and unconditional entropy dissipation for an implicit first order accurate in time, but higher order accurate in space, KKT-LDG discretizations are proved, which provides a first theoretical analysis of the KKT limiter. Finally, numerical results demonstrate the higher order accuracy and entropy dissipation of the KKT-DIRK-LDG discretizations for problems requiring a positivity limiter.
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