Seminar in Numerical Analysis: Jaap van der Vegt (Universiteit Twente)

Positivity Preserving Limiters for Time-Implicit Higher Order Accurate Discontinuous Galerkin Discretizations

In the numerical solution of partial differential equations, it is frequently necessary to ensure that certain variables, e.g., density, pressure, or probability density distribution, remain within strict bounds. Strict observation of these bounds is crucial, otherwise unphysical solutions will be obtained that might result in the failure of the numerical algorithm. Bounds on certain variables are generally ensured in discontinuous Galerkin (DG) discretizations using positivity preserving limiters, which locally modify the solution to ensure that the constraints are satisfied, while preserving higher order accuracy. In practice this approach is mostly limited to DG discretizations combined with explicit time integration methods. The combination of (positivity preserving) limiters in DG discretizations and implicit time integration methods results, however, in serious problems. Many positivity preserving limiters are not easy to apply in time-implicit DG discretizations and have a non-smooth formulation, which hampers the use of standard Newton methods to solve the nonlinear algebraic equations resulting from the time-implicit DG discretization. This often results in poor convergence.

In this presentation, we will discuss a different approach to ensure that a higher order accurate DG solution satisfies the positivity constraints. Instead of using a limiter, we impose the positivity constraints directly on the algebraic equations resulting from a higher order accurate time-implicit DG discretization using techniques from mathematical optimization theory.  This approach ensures that the positivity constraints are satisfied and does not affect the higher order accuracy of the time-implicit DG discretization. The resulting algebraic equations are then solved using a specially designed semi-smooth Newton method that is well suited to deal with the resulting nonlinear complementarity problem. We will demonstrate the algorithm on several parabolic model problems.

For further information about the seminar, please visit this webpage.

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