Abstract
In this contribution we derive and analyze a new numerical method for kinetic equations based on a variable transformation of the moment approximation. Classical minimum-entropy moment closures are a class of reduced models for kinetic equations that conserve many of the fundamental physical properties of solutions. However, their practical use is limited by their high computational cost, as an optimization problem has to be solved for every cell in the space-time grid. In addition, implementation of numerical solvers for these models is hampered by the fact that the optimization problems are only well-defined if the moment vectors stay within the realizable set. For the same reason, further reducing these models by, e.g., reduced-basis methods is not a simple task. Our new method overcomes these disadvantages of classical approaches. The transformation is performed on the semi-discretized level which makes them applicable to a wide range of kinetic schemes and replaces the nonlinear optimization problems by inversion of the positive-definite Hessian matrix. As a result, the new scheme gets rid of the realizability-related problems. Moreover, a discrete entropy law can be enforced by modifying the time stepping scheme. Our numerical experiments demonstrate that our new method is often several times faster than the standard optimization-based scheme.
Highlights
IntroductionThe regularized problem admits a solution 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, can be controlled by the regularization parameter)
Kinetic equations play an important role in many physical applications
This is usually done by performing a Galerkin projection of the original kinetic equation to the linear span of the weight functions
Summary
The regularized problem admits a solution 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, can be controlled by the regularization parameter). This approach still requires the solution of the (regularized) minimum entropy problem in each cell of the space-time grid.
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