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Register a new userExplicit and implicit error inhibiting schemes with post-processing
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Added by
Sigal Gottlieb
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Efficient high order numerical methods for evolving the solution of an ordinary differential equation are widely used. The popular Runge--Kutta methods, linear multi-step methods, and more broadly general linear methods, all have a global error that is completely determined by analysis of the local truncation error. In prior work in we investigated the interplay between the local truncation error and the global error to construct {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. In this work we extend our error inhibiting framework to include a broader class of time-discretization methods that allows an exact computation of the leading error term, which can then be post-processed to obtain a solution that is two orders higher than expected from truncation error analysis. We define sufficient conditions that result in a desired form of the error and describe the construction of the post-processor. A number of new explicit and implicit methods that have this property are given and tested on a variety of ordinary and partial differential equation. We show that these methods provide a solution that is two orders higher than expected from truncation error analysis alone.
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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.
High order methods are often desired for the evolution of ordinary differential equations, in particular those arising from the semi-discretization of partial differential equations. In prior work in we investigated the interplay between the local truncation error and the global error to construct error inhibiting general linear methods (GLMs) that control the accumulation of the local truncation error over time. Furthermore we defined sufficient conditions that allow us to post-process the final solution and obtain a solution that is two orders of accuracy higher than expected from truncation error analysis alone. In this work we extend this theory to the class of two-derivative GLMs. We define sufficient conditions that control the growth of the error so that the solution is one order higher than expected from truncation error analysis, and furthermore define the construction of a simple post-processor that will extract an additional order of accuracy. Using these conditions as constraints, we develop an optimization code that enables us to find explicit two-derivative methods up to eighth order that have favorable stability regions, explicit strong stability preserving methods up to seventh order, and A-stable implicit methods up to fifth order. We numerically verify the order of convergence of a selection of these methods, and the total variation diminishing performance of some of the SSP methods. We confirm that the methods found perform as predicted by the theory developed herein.
Systems driven by multiple physical processes are central to many areas of science and engineering. Time discretization of multiphysics systems is challenging, since different processes have different levels of stiffness and characteristic time scales. The multimethod approach discretizes each physical process with an appropriate numerical method; the methods are coupled appropriately such that the overall solution has the desired accuracy and stability properties. The authors developed the general-structure additive Runge-Kutta (GARK) framework, which constructs multimethods based on Runge-Kutta schemes. This paper constructs the new GARK-ROS/GARK-ROW families of multimethods based on linearly implicit Rosenbrock/Rosenbrock-W schemes. For ordinary differential equation models, we develop a general order condition theory for linearly implicit methods with any number of partitions, using exact or approximate Jacobians. We generalize the order condition theory to two-way partitioned index-1 differential-algebraic equations. Applications of the framework include decoupled linearly implicit, linearly implicit/explicit, and linearly implicit/implicit methods. Practical GARK-ROS and GARK-ROW schemes of order up to four are constructed.
Many complex applications require the solution of initial-value problems where some components change fast, while others vary slowly. Multirate schemes apply different step sizes to resolve different components of the system, according to their dynamics, in order to achieve increased computational efficiency. The stiff components of the system, fast or slow, are best discretized with implicit base methods in order to ensure numerical stability. To this end, linearly implicit methods are particularly attractive as they solve only linear systems of equations at each step. This paper develops the Multirate GARK-ROS/ROW (MR-GARK-ROS/ROW) framework for linearly-implicit multirate time integration. The order conditions theory considers both exact and approximative Jacobians. The effectiveness of implicit multirate methods depends on the coupling between the slow and fast computations; an array of efficient coupling strategies and the resulting numerical schemes are analyzed. Multirate infinitesimal step linearly-implicit methods, that allow arbitrarily small micro-steps and offer extreme computational flexibility, are constructed. The new unifying framework includes existing multirate Rosenbrock(-W) methods as particular cases, and opens the possibility to develop new classes of highly effective linearly implicit multirate integrators.
We consider the development of high order space and time numerical methods based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic systems with relaxation. More specifically, we consider hyperbolic balance laws in which the convection and the source term may have very different time and space scales. As a consequence the nature of the asymptotic limit changes completely, passing from a hyperbolic to a parabolic system. From the computational point of view, standard numerical methods designed for the fluid-dynamic scaling of hyperbolic systems with relaxation present several drawbacks and typically lose efficiency in describing the parabolic limit regime. In this work, in the context of Implicit-Explicit linear multistep methods we construct high order space-time discretizations which are able to handle all the different scales and to capture the correct asymptotic behavior, independently from its nature, without time step restrictions imposed by the fast scales. Several numerical examples confirm the theoretical analysis.
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