On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators

Abstract

Many important initial value problems have the property that energy is nonincreasing in time. Energy stable methods, also referred to as strongly stable methods, guarantee the same property discretely. We investigate requirements for conditional energy stability of explicit Runge-Kutta methods for nonlinear or non-autonomous problems. We provide both necessary and sufficient conditions for energy stability over these classes of problems. Examples of conditionally energy stable schemes are constructed and an example is given in which unconditional energy stability is obtained with an explicit scheme. IntroductionEver since the construction of numerical methods for ordinary and (time-dependent) partial differential equations (ODEs and PDEs, respectively), their stability has been an important and active topic of research. Monotonicity, meaning that the norm of the solution is bounded by its initial value, is a particularly exacting stability property. For equations, such as parabolic PDEs, that contain a significant amount of dissipation, any reasonable numerical method will typically preserve monotonicity under an appropriate time step restriction. In contrast, for non-dissipative problems such as hyperbolic PDEs (and their slightly dissipative semidiscretizations), common time discretizations may not preserve monotonicity under any finite step size.The energy method is an effective tool to get stability estimates, e.g. for hyperbolic PDEs [12,19]. Using summation by parts operators [9,34], these can be transferred efficiently to the semidiscrete level for many different kinds of schemes [11,20,21,28]. However, applying the same approach in time yields implicit methods [2,10,22,25]. Classical nonlinearly stable methods, such as algebraically stable Runge-Kutta methods, are also implicit. For hyperbolic problems, such implicit methods are usually less efficient than explicit ones. It is possible to obtain conditional energy stability with explicit methods by using modifications that go outside the class of Runge-Kutta methods; e.g. projection methods [5,6,13] and relaxation Runge-Kutta schemes [18,29].Nevertheless, it is interesting to know what can be achieved within the class of explicit Runge-Kutta methods without modifications. In this setting, results have been obtained for problems that include a certain amount of dissipation [8,16]. Recently this topic has again attracted the interest of researchers and several results (using the term strong stability) for linear, time-independent operators have been discovered [27,32,33]. Nonlinear problems have been investigated in [24], where many non-existence results for energy stable and strong stability 1 arXiv:1909.13215v1 [math.NA]

Energy Stability of Explicit Runge--Kutta Methods for Nonautonomous or Nonlinear Problems

“…Typically, conditional energy stability can be guaranteed for problems f (t, u) = Lu in this class under a time step restriction of the form \Delta t \leq C\| L\| - 1 , corresponding to a classical Courant--Friedrichs--Lewy criterion for discretizations of hyperbolic conservation laws [46,35,42,44]. Similar results have been obtained for some first order accurate schemes and autonomous semibounded nonlinear problems [29]. In the latter setting, the maximal time step is proportional to the inverse of the Lipschitz constant of the nonlinear right-hand side f of the ODE.…”

“…Recently this topic has again attracted the interest of researchers, and several results (using the term strong stability) for linear, time-independent operators have been discovered [35,42,44]. Nonlinear problems have been investigated in [29], where many nonexistence results for energy stable and strong stability preserving (SSP) methods of order two and greater have been proved. A first order accurate energy stable SSP method for autonomous problems has also been discovered therein.…”

Relaxation Runge--Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier--Stokes Equations

“…Stability/dissipation results for fully discrete schemes have mainly been limited to semidiscretizations including certain amounts of dissipation [31,32,49,71], linear equations [53,54,64,65,68], or fully implicit time integration schemes [7,10,11,26,36,39,48]. For explicit methods and general equations, there are negative experimental and theoretical results concerning energy/entropy stability [37,38,47,50].…”

A New Class of A Stable Summation by Parts Time Integration Schemes with Strong Initial Conditions

“…In contrast, the stability analysis of explicit time integration methods can use techniques similar to summation by parts, but the analysis is in general more complicated and restricted to sufficiently small time steps [36,44,46]. Since there are strict stability limitations for explicit methods, especially for nonlinear problems [30,31], an alternative to stable fully implicit methods is to modify less expensive (explicit or not fully implicit) time integration schemes to get the desired stability results [10,15,32,33,39,45].…”