Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity
Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity
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DOI：
10.1007/s42967-020-00098-y
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发表时间：
2019-12
影响因子：
1.6
通讯作者：
Zheng Sun;Chi-Wang Shu
中科院分区：
文献类型：
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作者：
Zheng Sun;Chi-Wang Shu
一个时间离散化方法称为强稳定（或单调），如果它的数值解的范数是非增的。虽然这一性质在各种情况下是可取的，许多显式Runge-Kutta（RK）方法可能无法保持它。在本文中，我们加强强稳定性修改的方法与超粘性，这是一个数值技术中常用的谱方法。我们的主要重点是强稳定性下的内积范数的线性问题可能非正常运营商。我们提出了两种方法稳定：修改的方法和过滤方法。修正方法是通过在半负算子中加入高阶超粘性项来实现的;滤波方法是通过求解具有小超粘性的扩散或色散问题来对解进行后处理。对于线性问题，大多数显式RK方法可以用任何一种方法稳定而不会出现精度退化。此外，我们证明了一个尖锐的边界（直到一个等号）扩散超粘性，以确保强稳定性。对于非线性问题，研究了一种滤波方法。数值例子与线性非正常常微分方程系统和间断Galerkin近似的守恒律进行验证我们的分析和测试的性能。
A time discretization method is called strongly stable (or monotone), if the norm of its numerical solution is nonincreasing. Although this property is desirable in various of contexts, many explicit Runge-Kutta (RK) methods may fail to preserve it. In this paper, we enforce strong stability by modifying the method with superviscosity, which is a numerical technique commonly used in spectral methods. Our main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal operators. We propose two approaches for stabilization: the modified method and the filtering method. The modified method is achieved by modifying the semi-negative operator with a high order superviscosity term; the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small superviscosity. For linear problems, most explicit RK methods can be stabilized with either approach without accuracy degeneration. Furthermore, we prove a sharp bound (up to an equal sign) on diffusive superviscosity for ensuring strong stability. For nonlinear problems, a filtering method is investigated. Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.