General relaxation methods for initial-value problems with application to multistep schemes

Ranocha, Hendrik; Lóczi, Lajos; Ketcheson, David I.

doi:10.1007/s00211-020-01158-4

Cited by 39 publications

(38 citation statements)

References 59 publications

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“…Herein we employ one-step integration methods and we enforce the conservation property discretely in time, so that ( ) = ( −1 ) = ( 0 ). We can achieve this by combining our conservative spatial discretizations with relaxation methods in time [53,84,85,89]. We start with a Runge-Kutta method where we use the shorthand := ( + Δ , ).…”

Section: Relaxation Methods In Timementioning

confidence: 99%

“…For some dispersive wave problems, the value of might change over time due to boundary conditions or the presence of dissipative terms; in this case relaxation methods can also be used to improve the accuracy of the time evolution of [53,89]. For more details regarding the properties of relaxation methods, including multistep relaxation methods, we refer the reader to [85].…”

Section: Relaxation Methods In Timementioning

confidence: 99%

“…To transfer the semidiscrete conservation results to fully-discrete schemes, the recent relaxation approach is used [53,[83][84][85]89]. First ideas for such techniques date back to [92,93] and [32, pp.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

Ranocha¹,

Mitsotakis

Ketcheson

2021

CICP

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We develop a general framework for designing conservative numerical methods based on summation by parts operators and split forms in space, combined with relaxation Runge-Kutta methods in time. We apply this framework to create new classes of fully-discrete conservative methods for several nonlinear dispersive wave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm, Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizations conserve all linear invariants and one nonlinear invariant for each system. The spatial semidiscretizations include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentially explicit, using relaxation Runge-Kutta methods. We implement some specific schemes from among the derived classes, and demonstrate their favorable properties through numerical tests.

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Section: Relaxation Methods In Timementioning

confidence: 99%

Section: Relaxation Methods In Timementioning

confidence: 99%

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A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

Ranocha¹,

Mitsotakis

Ketcheson

2021

CICP

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“…The disadvantage of the explicit Runge-Kutta methods tested in the previously mentioned works is that they do not conserve the energy functional. Therefore, the fully-discrete systems proposed so far for the numerical solution of the system (1) are not conservative with the exception of the very recent work [33], where appropriate collocation methods were considered for the spatial discretisation combined with recently proposed relaxation Runge-Kutta methods for integration in time [25,34,32].…”

Section: Introductionmentioning

confidence: 99%

A Conservative Fully Discrete Numerical Method for the Regularized Shallow Water Wave Equations

Mitsotakis

Ranocha²,

Ketcheson

et al. 2021

SIAM J. Sci. Comput.

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The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fullydiscrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge-Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to standard Galerkin methods. E(t; η, u) := 1 2 I gη 2 + (D + η)u 2 dx, (3)

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“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Ranocha

Nordström

2021

J Sci Comput

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Since integration by parts is an important tool when deriving energy or entropy estimates for differential equations, one may conjecture that some form of summation by parts (SBP) property is involved in provably stable numerical methods. This article contributes to this topic by proposing a novel class of A stable SBP time integration methods which can also be reformulated as implicit Runge-Kutta methods. In contrast to existing SBP time integration methods using simultaneous approximation terms to impose the initial condition weakly, the new schemes use a projection method to impose the initial condition strongly without destroying the SBP property. The new class of methods includes the classical Lobatto IIIA collocation method, not previously formulated as an SBP scheme. Additionally, a related SBP scheme including the classical Lobatto IIIB collocation method is developed.

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General relaxation methods for initial-value problems with application to multistep schemes

Cited by 39 publications

References 59 publications

A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

A Conservative Fully Discrete Numerical Method for the Regularized Shallow Water Wave Equations

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

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