Order reduction and how to avoid it when explicit Runge–Kutta–Nyström methods are used to solve linear partial differential equations

Alonso‐Mallo, Isaías; Cano, Begoña; Moreta, M. J.

doi:10.1016/j.cam.2004.07.021

Cited by 16 publications

(10 citation statements)

References 19 publications

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“…However, we will be interested in regular solutions in space and time for which the solution and many spatial derivatives are negligible outside a bounded domain. Our experience on exponential methods for other types of problems and on the study of order reduction with more classical methods leads us to suspect that, in such a case, we can get an order of accuracy as high as we want with Lawson methods (the classical order of the underlying methods.) Although it is not an aim to prove that in this article, that is in fact observed in the numerical experiments.…”

Section: Introductionmentioning

confidence: 99%

Projected explicit lawson methods for the integration of Schrödinger equation

Cano

González-Pachón

2014

Numerical Methods Partial

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In this article, it is proved that explicit Lawson methods, when projected onto one of the invariants of nonlinear Schrödinger equation (norm) are also automatically projected onto another invariant (momentum) for many solutions. As this procedure is very cheap and geometric because two invariants are conserved, it offers an efficient tool to integrate some solutions of this equation till long times. On the other hand, we show a detailed study on the numerical performance of these methods against splitting ones, with fixed and variable stepsize implementation.

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Section: Introductionmentioning

confidence: 99%

Projected explicit lawson methods for the integration of Schrödinger equation

Cano

González-Pachón

2014

Numerical Methods Partial

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“…Finally a simple theoretical analysis explains this reduced accuracy. While order-reduction in Runge-Kutta methods for partial differential equations is well understood going back to [18] and also applies to Runge-Kutta Nystrom Methods [1], the phenomenon observed here is not related to the boundary conditions ( as zero Dirichlet conditions are used), but arises from the differentiation and interpolation errors that occur when moving from particles to the grid and back again. This also matches the results seen for displacement error by the author [5] where the Stormer-Verlet Method provides much better energy conservation the Symplectic Euler method but the displacement errors are very similar.…”

Section: Introductionmentioning

confidence: 93%

Symplectic Time Integration Methods for the Material Point Method, Experiments, Analysis and Order Reduction

Berzins¹

2021

14th WCCM-ECCOMAS Congress

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The provision of appropriate time integration methods for the Material Point Method (MPM) involves considering stability, accuracy and energy conservation. A class of methods that addresses many of these issues are the widely-used symplectic time integration methods. Such methods have good conservation properties and have the potential to achieve high accuracy. In this work we build on the work in [5] and consider high order methods for the time integration of the Material Point Method. The results of practical experiments show that while high order methods in both space and time have good accuracy initially, unless the problem has relatively little particle movement then the accuracy of the methods for later time is closer to that of low order methods. A theoretical analysis explains these results as being similar to the stage error found in Runge Kutta methods, though in this case the stage error arises from the MPM differentiations and interpolations from particles to grid and back again, particularly in cases in which there are many grid crossings.

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“…These stiff systems arise in a wide range of applications. In particular, when some second-order in time partial differential equations are discretized in space, we obtain problems like (1.1) which are arbitrarily stiff [1].…”

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confidence: 99%

Stable Runge–Kutta–Nyström methods for dissipative stiff problems

2006

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The definition of stability for Runge-Kutta-Nyström methods applied to stiff second-order in time problems has been recently revised, proving that it is necessary to add a new condition on the coefficients in order to guarantee the stability. In this paper, we study the case of second-order in time problems in the nonconservative case. For this, we construct an R-stable Runge-Kutta-Nyström method with two stages satisfying this condition of stability and we show numerically the advantages of this new method.Keywords Runge-Kutta-Nyström methods · stability AMS subject classifications 65L06 · 65L20 PreliminariesWe consider a stiff second-order in time initial value problem(1.1)

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Order reduction and how to avoid it when explicit Runge–Kutta–Nyström methods are used to solve linear partial differential equations

Cited by 16 publications

References 19 publications

Projected explicit lawson methods for the integration of Schrödinger equation

Projected explicit lawson methods for the integration of Schrödinger equation

Symplectic Time Integration Methods for the Material Point Method, Experiments, Analysis and Order Reduction

Stable Runge–Kutta–Nyström methods for dissipative stiff problems

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