scholarly journals Multirate linearly-implicit GARK schemes

2021 ◽  
Author(s):  
Michael Günther ◽  
Adrian Sandu
Keyword(s):  
Numerical Stability ◽  
Initial Value Problems ◽  
Time Integration ◽  
Order Conditions ◽  
Systems Of Equations ◽  
Implicit Methods ◽  
Initial Value ◽  
Multirate Time Integration ◽  
Efficient Coupling ◽  
Linear Systems Of Equations

AbstractMany 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.

scholarly journals LISMA_HDS language for modeling heterogeneous dynamic systems

2021 ◽  
pp. 103-122
Author(s):  
Evgeny Popov ◽  
◽  
Yury Shornikov ◽  
Keyword(s):  
Dynamic Systems ◽  
Initial Value Problems ◽  
Method Of Lines ◽  
System Of Equations ◽  
Modeling Languages ◽  
Cauchy Problems ◽  
Systems Of Equations ◽  
Initial Value ◽  
Algebraic Systems ◽  
Different Types

Heterogeneous dynamic systems (HDS) simultaneously describe processes of different physical nature. Systems of this kind are typical for numerous applications. HDSs are characterized by the following features. They are often multimode or hybrid systems. In general, their modes are defined as initial value problems (Cauchy problems) for implicit differential-algebraic systems of equations. Due to the presence of heterogeneous dynamic components or processes evolving in both time and space, the dimension of the complete system of equations may be pretty high. In some cases, the system of equations has an internal structure, for instance, the differential-algebraic system of equations approximating a partial differential equation by the method of lines. An original huge system of equations can then be algorithmically rewritten in a compact form. Moreover, heterogeneous hybrid dynamical systems can generate events of qualitatively different types. Therefore one has to use different numerical event detection algorithms. Nowadays, HDSs are modeled and simulated in computer environments. The modeling languages widely used by engineers do not allow them to fully specify all the properties of the systems of this class. For instance, they do not include event typing constructs. That is why a declarative general-purpose modeling language named LISMA_HDS has been developed for the computer-aided modeling and ISMA simulation environment. The language takes into account all of the characteristic features of HDSs. It includes constructs for plain or algorithmic declaration of model constants, initial value problems for explicit differential-algebraic systems of equations, and initial guesses for variables. It also allows researchers to define explicit time events, modes and transitions between them upon the occurrence of events of different types, to use macros and implement event control. LISMA_HDS is defined by a generative grammar in an extended Backus-Naur form and semantic constraints. It is proved that the grammar belongs to the LL(2) subclass of context-free grammars.

scholarly journals Implicit Methods for Numerical Solution of Singular Initial Value Problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Md. Rahaman Habibur ◽  
M. Kamrul Hasan ◽  
Md. Ayub Ali ◽  
M. Shamsul Alam
Keyword(s):  
Singular Point ◽  
Numerical Solution ◽  
Initial Value Problems ◽  
Accurate Result ◽  
Implicit Method ◽  
Implicit Methods ◽  
Initial Value ◽  
Runge Kutta

AbstractVarious order of implicit method has been formulated for solving initial value problems having an initial singular point. The method provides better result than those obtained by used implicit formulae developed based on Euler and Runge-Kutta methods. Romberg scheme has been used for obtaining more accurate result.

scholarly journals A New Family of Phase-Fitted and Amplification-Fitted Runge-Kutta Type Methods for Oscillators

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Zhaoxia Chen ◽  
Xiong You ◽  
Xin Shu ◽  
Mei Zhang
Keyword(s):  
Initial Value Problems ◽  
Numerical Experiments ◽  
Order Conditions ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
New Methods ◽  
New Family ◽  
Algebraic Order ◽  
Runge Kutta ◽  
Phase Fitting

In order to solve initial value problems of differential equations with oscillatory solutions, this paper improves traditional Runge-Kutta (RK) methods by introducing frequency-depending weights in the update. New practical RK integrators are obtained with the phase-fitting and amplification-fitting conditions and algebraic order conditions. Two of the new methods have updates that are also phase-fitted and amplification-fitted. The linear stability and phase properties of the new methods are examined. The results of numerical experiments on physical and biological problems show the robustness and competence of the new methods compared to some highly efficient integrators in the literature.

Numerical relativity. I. The characteristic initial value problem

1982 ◽  
Vol 384 (1787) ◽  
pp. 427-454 ◽  
Keyword(s):  
Initial Data ◽  
Initial Value Problem ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Numerical Relativity ◽  
Hyperbolic Systems ◽  
Systems Of Equations ◽  
Initial Value ◽  
Hyperbolic Systems Of Equations ◽  
Characteristic Initial Value Problem

In this, the first of a series of papers on numerical relativity, the characteristic initial value problem is posed in a form suitable for numerical integration. It can be reduced to the solution of two initial value problems for sets of ordinary differential equations (on the initial surfaces) and the solution of two initial value problems for hyperbolic systems of equations, one linear, one quasilinear. The initial data may be specified freely. Subsequent papers will develop numerical solutions of Einstein’s equations with use of this formalism.

scholarly journals Model Problems in Numerical Stability Theory for Initial Value Problems

SIAM Review ◽  
1994 ◽  
Vol 36 (2) ◽  
pp. 226-257 ◽  
Author(s):  
A. M. Stuart ◽  
A. R. Humphries
Keyword(s):  
Stability Theory ◽  
Numerical Stability ◽  
Initial Value Problems ◽  
Initial Value
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