Runge–Kutta methods

Summary

In numerical analysis, the Runge–Kutta methods (ˈrʊŋəˈkʊtɑː ) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta. The Runge–Kutta method The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method". Let an initial value problem be specified as follows: : \frac{dy}{dt} = f(t, y), \quad y(t_0) = y_0. Here y is an unknown function (scalar or vector) of time t, which we would like to approximate; we are told that \frac{dy}{dt}, the rate at which y changes, is a function of t and of y itself. At the initial time t_0 the corresponding

Official source

https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods

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Élastodynamique linéaire pour problème géophysique et dispersion numérique

Julie Marthe Favre

Dans ce travail, on cherche à utiliser des méthodes capables de simuler la propagation des ondes élastiques en deux dimensions. Pour cela, nous avons utilisé les équations élastodynamiques linéaires et nous y avons appliqué des méthodes numériques. Ici, la méthode des éléments finis de Galerkin a été utilisée pour la semi-discrétisation en espace, et la discrétisation en temps a, quant à elle, été faite à l’aide des méthodes d’Euler rétrograde, Leapfrog et Runge Kutta 4. Les objectifs de ce travail consistaient tout d’abord à comparer ces méthodes en termes de convergence et de stabilité, puis d’appliquer la plus adaptée à la simulation des ondes. Cette comparaison a montré que la méthode de Runge Kutta 4 avait une condition de stabilité moins forte que celle de la méthode Leapfrog, et nous a finalement mené à choisir une méthode inconditionnellement stable, la méthode d’Euler rétrograde, pour simuler un scénario réel de propagation d’ondes sismiques en utilisant des conditions aux limites non-réfléchissantes. Les résultats concernant cette simulation semblent satisfaisants et pertinents, notamment du point de vue des conditions aux limites utilisées, qui sont apparues comme étant tout à fait appropriées. Enfin, le dernier objectif de ce travail était de procéder à une analyse de la dispersion numérique, qui nous a montré l’importance d’un haut degré polynômiale pour obtenir de bons résulats ainsi que l’influence de certains paramètres sur les erreurs obtenues.

2015

Mixed-precision explicit stabilized Runge-Kutta methods for single- and multi-scale differential equations

Matteo Croci, Giacomo Rosilho De Souza

Mixed-precision algorithms combine low-and high-precision computations in order to benefit from the performance gains of reduced-precision without sacrificing accuracy. In this work, we design mixed-precision Runge-Kutta-Chebyshev (RKC) methods, where high precision is used for accuracy, and low precision for stability. Generally speaking, RKC methods are low-order explicit schemes with a stability domain growing quadratically with the number of function evaluations. For this reason, most of the computational effort is spent on stability rather than accuracy purposes. In this paper, we show that a naive mixed-precision implementation of any Runge-Kutta scheme can harm the convergence order of the method and limit its accuracy, and we introduce a new class of mixed precision RKC schemes that are instead unaffected by this limiting behavior. We present three mixed-precision schemes: a first-and a second-order RKC method, and a first order multirate RKC scheme for multiscale problems. These schemes perform only the few function evaluations needed for accuracy (1 or 2 for first-and second-order methods respectively) in high precision, while the rest are performed in low precision. We prove that while these methods are essentially as cheap as their fully low-precision equivalent, they retain the stability and convergence order of their high-precision counterpart. Indeed, numerical experiments confirm that these schemes are as accurate as the corresponding high-precision method. (C) 2022 Elsevier Inc. All rights reserved.

ACADEMIC PRESS INC ELSEVIER SCIENCE2022

The aim of this project is to implement the Rosenbrock method ROS3P in the C++ Finite Element library LifeV for the solution of systems of ordinary differential equations arising in electrophysiology. In the domain of electrophysiology LifeV implements cardiac cell models which can require the resolution of stiff reaction-diffusion equations. To solve this system the use of implicit methods is required. We focus in particular on one step methods. Implicit methods need the solution of a non linear system at each time step. The Rosenbrock methods derive from the linearly implicit Runge Kutta methods and just requires the solution of s linear systems per time step, where s is the number of stages. Most of the implicit Runge Kutta methods suffer from order reduction when applied to parabolic problems. For this reason we have chosen the third order Rosenbrock method ROS3P, which satisfies additional order conditions to keep the same convergence order even when applied to non linear parabolic problems. In the first part of this work we derive the Rosenbrock methods from Runge Kutta methods, explaining the time adaptivity strategy and finally giving the algorithm of our implementation in LifeV. After we present the cardiac reaction-diffusion models, deriving the mono-domain model from the bi-domain model. In the following we explain the strategy that we adopt to solve these models and how we discretize them. Finally we show some numerical experiments obtained with our approach and compare our results with the ones given by other methods.

2013