Integrating Factor Methods as Exponential Integrators

Minchev, Borislav V.

doi:10.1007/11666806_43

Cited by 7 publications

(10 citation statements)

References 10 publications

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“…Under these constraints, for explicit method, by noting that c 1 = 0, U n1 = u n and U n𝜏,1 = u n𝜏 , we reformulate the exponential Rosenbrock methods ( 6) and (7) as…”

Section: Exponential Rosenbrock Methodsmentioning

confidence: 99%

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Convergence analysis of high‐order exponential Rosenbrock methods for nonlinear stiff delay differential equations

Zhan,

Fang

2023

Math Methods in App Sciences

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In this work, we extend previous research on exponential integrators for stiff semilinear delay differential equations to the nonlinear case. In addition to the stiffness, there are two new issues that should be handled properly: nonlinear term and delay term. For nonlinear problems, a badly chosen linearization can cause a severe step size restriction. In this work, we linearize the equation along the numerical solution in each step. For the delay term, the interpolation based on the numerical values at the mesh points rather than inner stage values is adopted to significantly reduce the number of stiff order conditions. We focus on the construction and convergence analysis of high‐order exponential Rosenbrock methods for nonlinear stiff delay differential equations. The main result of this paper is that under the framework of strongly continuous semigroup, the explicit exponential Rosenbrock method is proved to be stiffly convergent of order even if the order conditions of order hold in a weak form. Moreover, by pointing that there does not exist fifth‐order method with less than or equal to four stages, we present the construction of a fifth‐order method with five stages. Finally, numerical tests are carried out to validate the theoretical results and to demonstrate the superiority of high‐order methods.

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“…Under these constraints, for explicit method, by noting that c 1 = 0, U n1 = u n and U n𝜏,1 = u n𝜏 , we reformulate the exponential Rosenbrock methods ( 6) and (7) as…”

Section: Exponential Rosenbrock Methodsmentioning

confidence: 99%

“…By precisely treating the linear stiff part, exponential integrators have been quite successful in solving semilinear stiff ODEs [7,8]. In the aspect of convergent theory for stiff problems, to overcome the time step length restriction, we prefer methods that are stiffly convergent.…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of high‐order exponential Rosenbrock methods for nonlinear stiff delay differential equations

Zhan,

Fang

2023

Math Methods in App Sciences

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“…This equation is then solved by any numerical method (in this scheme it is CRK4), and the solution obtained is transformed back to the original variable. It is pertinent to note that the IF methods can be viewed as a subclass of exponential integrators (see [14] for discussion).…”

Section: Linearly Implicit Runge-kutta Methodsmentioning

confidence: 99%

Fourth-Order Numerical Methods for the Coupled Korteweg–de Vries Equations

Korostil¹,

Clarke²

2015

ANZIAM J.

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We compare six fixed-stepsize fourth-order numerical methods for a number of test problems described by a system of coupled Korteweg–de Vries equations. Particular attention is paid to the ability of these methods to preserve fixed points (solitary waves) and the invariants of the system, and establishing to what extent the conservation of integral invariants is indicative of the solution error for these methods.

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“…Exponential time differencing (ETD) schemes are known for a long time in computational electrodynamics; see [39] for a comprehensive review of ETD methods and their history. In this section, we describe the exponential time differencing fourth-order Runge-Kutta (ETD4RK) method which was proposed by Cox-Matthews [37].…”

Section: Exponential Time Differencingmentioning

confidence: 99%