Entropy dissipative higher order accurate positivity preserving time-implicit discretizations for nonlinear degenerate parabolic equations

TL;DR

This paper introduces entropy dissipative, positivity-preserving, higher order accurate implicit discretizations for nonlinear degenerate parabolic equations, combining LDG and DIRK methods with KKT limiters for stability and efficiency.

Contribution

It develops a novel combination of LDG, DIRK, and KKT limiters ensuring entropy dissipation, positivity, and higher order accuracy for complex parabolic equations.

Findings

Unconditional entropy dissipation of the implicit Euler-LDG scheme.

Positivity of solutions maintained via KKT limiter with mass conservation.

Numerical results confirm higher order accuracy and entropy dissipation.

Abstract

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.