Explicit stabilized multirate method for stiff differential equations
Assyr Abdulle, Marcus J. Grote, Giacomo Rosilho de Souza
备注：
With respect to the previous version: Added a new numerical experiment where the mRKC method (first-order) is compared against a second-order RKC method and implicit Euler method on a nonlinear problem
Stabilized Runge-Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized Runge-Kutta methods overcome the stringent stability condition of standard methods without sacrificing explicitness. However, when stiffness is only induced by a few components, as in the presence of spatially local mesh refinement, their efficiency deteriorates. To remove the crippling effect of a few severely stiff components on the entire system of differential equations, we derive a modified equation, whose stiffness solely depend on the remaining mildly stiff components. By applying stabilized Runge-Kutta methods to this modified equation, we then devise an explicit multirate Runge-Kutta-Chebyshev (mRKC) method whose stability conditions are independent of a few severely stiff components. Stability of the mRKC method is proved for a model problem, whereas its efficiency and usefulness are demonstrated through a series of numerical experiments.
Explicit stabilized multirate method for stiff stochastic differential equations
稳定化显式方法对于大型刚性随机微分方程（SDEs）系统是特别有效的，这是由于它们的扩展稳定域。然而，他们失去了他们的效率时，一个严重的刚度是由非常少的“快”的自由度，因为刚性和非刚性条款同时评估。”[10]“受启发。Abdulle，M. Grote和G. Rosilho de Souza，预印本（2020），arXiv：2006.00744]我们引入了一个随机修改的方程，其刚度仅取决于“慢”项。通过整合这个修改后的方程与稳定的显式计划，我们设计了一个多速率方法，克服了瓶颈所造成的一些严重的刚性条款和恢复的效率稳定计划的大型系统的非线性SDES。该方案不基于任何尺度分离的假设，因此，它是受雇于随机抛物型偏微分方程的空间离散的局部加密网格上产生的问题。该多速率格式具有强1/2阶和弱1阶，并证明了其稳定性。数值实验证实了格式的有效性和准确性。
A Weakly Asymptotic Preserving Low Mach Number Scheme for the Euler Equations of Gas Dynamics
RWTH-Aachen（亚琛工业大学）；Johannes Gutenberg-Universität Mainz（美因茨约翰内斯古腾堡大学）；Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育研究所特里凡得琅分校）；Universität Stuttgart（斯图加特大学）
