Since integration by parts is an important tool when deriving energy or entropy estimates for differential equations, one may conjecture that some form of summation by parts (SBP) property is involved in provably stable numerical methods. This article contributes to this topic by proposing a novel class of $A$ stable SBP time integration methods which can also be reformulated as implicit Runge-Kutta methods. In contrast to existing SBP time integration methods using simultaneous approximation terms to impose the initial condition weakly, the new schemes use a projection method to impose the initial condition strongly without destroying the SBP property. The new class of methods includes the classical Lobatto IIIA collocation method, not previously formulated as an SBP scheme. Additionally, a related SBP scheme including the classical Lobatto IIIB collocation method is developed.
翻译:由于各部分的整合是计算能量或对不同方程的微粒估计的一个重要工具,人们可以推测,某些部分财产(SBP)的某种形式加起来,涉及可以想象的稳定数字方法,本条通过提出一个新的类别,即稳定的SBP时间整合方法,也可以重新表述为隐含的龙格-库塔方法,对这一专题有帮助。与现有的SBP时间整合方法相比,新计划使用同步近似条件来弱化初始条件,采用预测方法在不破坏SBP财产的情况下强力实施初始条件。新的方法类别包括古典Lobatto IIIA合用法,而以前不是作为SBP计划拟订。此外,还制定了相关的SBP方案,包括传统的Lobatto IIIB合用法。