Runge-Kutta IMEX schemes for the Horizontally Explicit/Vertically Implicit (HEVI) solution of wave equations

Many operational weather forecasting centres use semi-implicit time-stepping schemes because of their good efficiency. However, as computers become ever more parallel, horizontally explicit solutions of the equations of atmospheric motion might become an attractive alternative due to the additional inter-processor communication of implicit methods. Implicit and explicit (IMEX) time-stepping schemes have long been combined in models of the atmosphere using semi-implicit, split-explicit or HEVI splitting. However, most studies of the accuracy and stability of IMEX schemes have been limited to the parabolic case of advection–diffusion equations. We demonstrate how a number of Runge–Kutta IMEX schemes can be used to solve hyperbolic wave equations either semi-implicitly or HEVI. A new form of HEVI splitting is proposed, UfPreb, which dramatically improves accuracy and stability of simulations of gravity waves in stratified flow. As a consequence it is found that there are HEVI schemes that do not lose accuracy in comparison to semi-implicit ones. The stability limits of a number of variations of trapezoidal implicit and some Runge–Kutta IMEX schemes are found and the schemes are tested on two vertical slice cases using the compressible Boussinesq equations split into various combinations of implicit and explicit terms. Some of the Runge–Kutta schemes are found to be beneficial over trapezoidal, especially since they damp high frequencies without dropping to first-order accuracy. We test schemes that are not formally accurate for stiff systems but in stiff limits (nearly incompressible) and find that they can perform well. The scheme ARK2(2,3,2) performs the best in the tests

Asymptotic Preserving time-discretization of optimal control problems for the Goldstein-Taylor model

We consider the development of implicit-explicit time integration schemes for optimal control problems governed by the Goldstein-Taylor model. In the diffusive scaling this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman-Enskog type limiting procedure. For the class of stiffly accurate implicit-explicit Runge-Kutta methods (IMEX) the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior

A class of high-order Runge-Kutta-Chebyshev stability polynomials

The analytic form of a new class of factorized Runge-Kutta-Chebyshev (FRKC) stability polynomials of arbitrary order $N$ is presented. Roots of FRKC stability polynomials of degree $L=MN$ are used to construct explicit schemes comprising $L$ forward Euler stages with internal stability ensured through a sequencing algorithm which limits the internal amplification factors to $\sim L^2$. The associated stability domain scales as $M^2$ along the real axis. Marginally stable real-valued points on the interior of the stability domain are removed via a prescribed damping procedure. By construction, FRKC schemes meet all linear order conditions; for nonlinear problems at orders above 2, complex splitting or Butcher series composition methods are required. Linear order conditions of the FRKC stability polynomials are verified at orders 2, 4, and 6 in numerical experiments. Comparative studies with existing methods show the second-order unsplit FRKC2 scheme and higher order (4 and 6) split FRKCs schemes are efficient for large moderately stiff problems.Comment: 24 pages, 5 figures. Accepted for publication in Journal of Computational Physics, 22 Jul 2015. Revise

Unconditional Stability for Multistep ImEx Schemes: Theory

This paper presents a new class of high order linear ImEx multistep schemes with large regions of unconditional stability. Unconditional stability is a desirable property of a time stepping scheme, as it allows the choice of time step solely based on accuracy considerations. Of particular interest are problems for which both the implicit and explicit parts of the ImEx splitting are stiff. Such splittings can arise, for example, in variable-coefficient problems, or the incompressible Navier-Stokes equations. To characterize the new ImEx schemes, an unconditional stability region is introduced, which plays a role analogous to that of the stability region in conventional multistep methods. Moreover, computable quantities (such as a numerical range) are provided that guarantee an unconditionally stable scheme for a proposed implicit-explicit matrix splitting. The new approach is illustrated with several examples. Coefficients of the new schemes up to fifth order are provided.Comment: 33 pages, 7 figure

Multi-Rate Runge-Kutta-Chebyshev Time Stepping for Parabolic Equations on Adaptively Refined Meshes

In this thesis, we develop an explicit multi-rate time stepping method for solving parabolic equations on a one dimensional adaptively refined mesh. Parabolic equations are characterized by their stiffness and as a result are usually solved using implicit time stepping schemes [16]. However, implicit schemes have the disadvantage that they can be expensive in higher dimensions or complicated to implement on adaptive or otherwise non-uniform meshes. Moreover, for coupled systems of parabolic equations, it can be difficult to achieve the expected order of accuracy without using sophisticated operator splitting techniques. For these reasons, we seek to exploit the properties of stabilized explicit methods for parabolic equations. In particular, we use the Runge-Kutta-Chebyshev (RKC) methods, a family of explicit Runge-Kutta methods, with numerical stability regions that extend far into the left half plane [12, 15, 21, 22, 26, 27, 28]. A central goal of this thesis is to use a second order RKC scheme to numerically solve parabolic equations on a one dimensional adaptively refined finite volume mesh. To make our implementation efficient, we design a time stepping algorithm in which time step sizes are chosen to respect the local mesh widths. This time stepping process requires communication between the RKC stages on different refinement levels. By linearly interpolating in time between the stage values, we obtain the ghost cell values for the finite volume scheme on each level. To our knowledge, this approach to adaptively refining in time, commonly referred to as a multi-rate time stepping strategy, combined with RKC time stepping method has not been previously implemented. We develop our multi-rate algorithm on a one dimensional statically refined mesh using the second order finite volume scheme to numerically solve the heat or diffusion equation on each grid stored in a hierarchy of meshes. Using the method of manufactured solutions , we demonstrate that our method is second order accurate, and for our test problem, the multi-rate scheme requires only about 20% of the computational work required by the uniformly refined mesh at the same resolution. The algorithm we develop manages the time stepping between the refinement levels only, and so extends directly to higher dimensional problems. Future work in this direction includes applying the new multi-rate RKC time stepping scheme to biological pattern formations or crystal growth in the 2D ForestClaw code [7] on parallel machines