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Register a new userError Estimation for Multi-Stage Runge-Kutta IMEX Schemes
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Implicit-Explicit (IMEX) schemes are widely used for time integration methods for approximating solutions to a large class of problems. In this work, we develop accurate a posteriori error estimates of a quantity of interest for approximations obtained from multi-stage IMEX schemes. This is done by first defining a finite element method that is nodally equivalent to an IMEX scheme, then using typical methods for adjoint-based error estimation. The use of a nodally equivalent finite element method allows a decomposition of the error into multiple components, each describing the effect of a different portion of the method on the total error in a quantity of interest.
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In this work we present a class of high order unconditionally strong stability preserving (SSP) implicit multi-derivative Runge--Kutta schemes, and SSP implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order $p>2$ is unique among SSP methods, and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge--Kutta schemes of order up to $p=4$, and IMEX Runge--Kutta schemes of order up to $p=3$. For the multi-derivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar-Gross-Krook (BGK) kinetic equation. We present numerical results to support the theoretical results, on a variety of problems.
High order implicit-explicit (IMEX) methods are often desired when evolving the solution of an ordinary differential equation that has a stiff part that is linear and a non-stiff part that is nonlinear. This situation often arises in semi-discretization of partial differential equations and many such IMEX schemes have been considered in the literature. The methods considered usually have a a global error that is of the same order as the local truncation error. More recently, methods with global errors that are one order higher than predicted by the local truncation error have been devised (by Kulikov and Weiner, Ditkowski and Gottlieb). In prior work we investigated the interplay between the local truncation error and the global error to construct explicit and implicit {em error inhibiting schemes} that control the accumulation of the local truncation error over time, resulting in a global error that is one order higher than expected from the local truncation error, and which can be post-processed to obtain a solution which is two orders higher than expected. In this work we extend our error inhibiting with post-processing framework introduced in our previous work to a class of additive general linear methods with multiple steps and stages. We provide sufficient conditions under which these methods with local truncation error of order p will produce solutions of order (p+1), which can be post-processed to order (p+2), and describe the construction of one such post-processor. We apply this approach to obtain implicit-explicit (IMEX) methods with multiple steps and stages. We present some of our new IMEX methods and show their linear stability properties, and investigate how these methods perform in practice on some numerical test cases.
We present unconditionally energy stable Runge-Kutta (RK) discontinuous Galerkin (DG) schemes for solving a class of fourth order gradient flows. Our algorithm is geared toward arbitrarily high order approximations in both space and time, while energy dissipation remains preserved without imposing any restriction on time steps and meshes. We achieve this in two steps. First, taking advantage of the penalty free DG method introduced by Liu and Yin [J Sci. Comput. 77:467--501, 2018] for spatial discretization, we reformulate an extended linearized ODE system by the energy quadratization (EQ) approach. Second, we apply an s-stage algebraically stable RK method for temporal discretization. The resulting fully discrete DG schemes are linear and unconditionally energy stable. In addition, we introduce a prediction-correction procedure to improve both the accuracy and stability of the scheme. We illustrate the effectiveness of the proposed schemes by numerical tests with benchmark problems.
We construct a family of embedded pairs for optimal strong stability preserving explicit Runge-Kutta methods of order $2 leq p leq 4$ to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction, the goals include non-defective methods, large region of absolute stability, and optimal error measurement as defined in [5,19]. The new family of embedded pairs offer the ability for strong stability preserving (SSP) methods to adapt by varying the step-size based on the local error estimation while maintaining their inherent nonlinear stability properties. Through several numerical experiments, we assess the overall effectiveness in terms of precision versus work while also taking into consideration accuracy and stability.
We categorify the RK family of numerical integration methods (explicit and implicit). Namely we prove that if a pair of ODEs are related by an affine map then the corresponding discrete time dynamical systems are also related by the map. We show that in practice this works well when the pairs of related ODEs come from the coupled cell networks formalism and, more generally, from fibrations of networks of manifolds.
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