A Regularized Entropy-Based Moment Method for Kinetic Equations

Abstract

The entropy-based moment method is a well-known discretization for the velocity variable in kinetic equations which has many desirable theoretical properties but is difficult to implement with high-order numerical methods. The regularized entropy-based moment method was recently introduced to remove one of the main challenges in the implementation of the entropy-based moment method, namely the requirement of the realizability of the numerical solution. In this work we use the method of relative entropy to prove the convergence of the regularized method to the original method as the regularization parameter goes to zero and give convergence rates. Our main assumptions are the boundedness of the velocity domain and that the original moment solution is Lipschitz continuous in space and bounded away from the boundary of realizability. We provide results from numerical simulations showing that the convergence rates we prove are optimal.

A realizable filtered intrusive polynomial moment method

“…A modification which achieves this is the regularization proposed in [38,33], in which the equality constraints are exchanged with a penalty term in the objective function. Specifically, the primal optimization problem (16) defining the ansatz is replaced by…”

“…One remaining question is how to choose the regularization parameter η. A discussion of this issue can be found in [33], where the authors choose η according to the Morozov discrepancy principle, when interpreting the numerical errors as noise. Hence, η is chosen such that the effect of regularization does not dominate the numerical error.…”

“…The rest of this paper is structured as follows: First we discuss how applying a standard filter to a realizable moment vector does not necessarily preserve its realizability and propose an alternate filter which does preserve realizability in Section 2. Then in the following Section 2.3, we apply the regularization technique from [33] to the optimization problem, thus making it feasible even for nonrealizable moment vectors and allowing the use of standard filters. We discuss our implementation in Sections 3 and present numerical results in Section 4.…”

A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations

“…We will always assume that the ansatz exactly reproduces the moments, i.e., b ψu,b = u. Note that this may not be fulfilled by regularized moment approximations as regarded, e.g., in [2].…”

“…Another approach to fix the realizability issues is to introduce a regularization of the optimization problem [2]. The regularized problem admits a solution also for moments vectors that are not realizable and maintains most of the desirable properties of the original problem, at the cost of an additional approximation error (which, however, can be controlled by the regularization parameter).…”