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Register a new userTwo-derivative 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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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.
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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.
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.
The context of this paper is the simulation of parameter-dependent partial differential equations (PDEs). When the aim is to solve such PDEs for a large number of parameter values, Reduced Basis Methods (RBM) are often used to reduce computational costs of a classical high fidelity code based on Finite Element Method (FEM), Finite Volume (FVM) or Spectral methods. The efficient implementation of most of these RBM requires to modify this high fidelity code, which cannot be done, for example in an industrial context if the high fidelity code is only accessible as a black-box solver. The Non Intrusive Reduced Basis method (NIRB) has been introduced in the context of finite elements as a good alternative to reduce the implementation costs of these parameter-dependent problems. The method is efficient in other contexts than the FEM one, like with finite volume schemes, which are more often used in an industrial environment. In this case, some adaptations need to be done as the degrees of freedom in FV methods have different meenings. At this time, error estimates have only been studied with FEM solvers. In this paper, we present a generalisation of the NIRB method to Finite Volume schemes and we show that estimates established for FEM solvers also hold in the FVM setting. We first prove our results for the hybrid-Mimetic Finite Difference method (hMFD), which is part the Hybrid Mixed Mimetic methods (HMM) family. Then, we explain how these results apply more generally to other FV schemes. Some of them are specified, such as the Two Point Flux Approximation (TPFA).
We derive a residual based a-posteriori error estimate for the outer normal derivative of approximations to Poissons problem. By analyzing the solution of the adjoint problem, we show that error indicators in the bulk may be defined to be of higher order than those close to the boundary, which lead to more economic meshes. The theory is illustrated with some numerical examples.
We introduce and analyze a class of Galerkin-collocation discretization schemes in time for the wave equation. Its conceptual basis is the establishment of a direct connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex linear algebraic systems. Continuously differentiable in time discrete solutions are obtained by the application of a special quadrature rule involving derivatives. Optimal order error estimates are proved for fully discrete approximations based on the Galerkin-collocation approach. Further, the concept of Galerkin-collocation approximation is extended to twice continuously differentiable in time discrete solutions. A direct connection between the two families by a computationally cheap post-processing is presented. The error estimates are illustrated by numerical experiments.
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