论文标题
一种新的基于可变量的离散方法，用于线性动力学方程的最小熵矩近似值
A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations
论文作者
论文摘要
在此贡献中，我们根据矩近似的可变变换来得出动力学方程的新数值方法。经典的最小值 - 内部矩闭合是一类减少动力学方程的模型，这些模型保留了许多溶液的基本物理特性。但是，它们的实际使用受到高计算成本的限制，因为必须在时空网格中为每个单元格解决一个优化问题。此外，这些模型的数值求解器的实现会受到这样一个事实的阻碍：仅当矢量留在可实现的集合中时，优化问题才明确定义。出于同样的原因，通过减少基本方法进一步降低这些模型并不是一个简单的任务。我们的新方法克服了经典方法的这些缺点。转换是在半差异的水平上执行的，这使它们适用于各种动力学方案，并通过逆转阳性黑森矩阵来代替非线性优化问题。结果，新方案摆脱了与可靠性相关的问题。此外，可以通过修改时间步进方案来执行离散的熵法。我们的数值实验表明，我们的新方法通常比基于标准优化的方案快几倍。
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.
