New Runge-Kutta Method for Stiff Oscillatory Problems with Two Frequencies

Fang, Yonglei; Ming, Qinghe

doi:10.1063/1.3241628

Cited by 2 publications

(5 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is known that exponential Runge-Kutta methods have the unique advantage to solve highly oscillatory systems, the coefficients of these exponential integrators are related to the matrix exponentials, so that the implementation of ERK methods heavily depends on the evaluations of matrix exponentials. In order to reduce computational cost, two new classes of explicit ERK integrators are formulated in recent papers [21,35]. As a sequel to our work, implicit ERK methods are studied in this paper.…”

Section: Discussionmentioning

confidence: 99%

“…However, the w s (z) and ws (z) are independent of matrix-valued exponentials. We have presented that the same order of these two new exponential methods share the same w s (z) or ws (z), and w s (z) is different from ws (z) when p ≥ 3 in [21,35]. It is obvious that the SVERK and MVERK methods reduce to the standard RK methods when M → 0, and these methods can exactly integrate the first-order homogeneous linear system y ′ (t) = −M y(t), y(0) = y 0 ,…”

Section: The Symplectic Conditions For Two New Classes Of Erk Methodsmentioning

confidence: 99%

“…1 (b) and (c), and we also plot the stability region of the explicit second-order MVERK method with two stages (see, e.g. [35]) in Fig. 1 (a).…”

Section: Linear Stabilitymentioning

confidence: 99%

“…[20,32]), Schrödinger equations (see, e.g. [3,6,35]) and KdV equations (see, e.g. [36]) can be converted into (1) with appropriate spatial discretization.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, our study is related to the classical order (nonstiff). We have presented that two new kinds of explicit ERK methods up to order four, which reduce to classical Runge-Kutta (RK) methods once M → 0 and their coefficients are independent of matrix-valued exponentials [21,35]. In what follows, we will study the implicit ERK methods.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Cost-reduction implicit exponential Runge-Kutta methods for highly oscillatory systems

Hu,

Wang,

Wang

et al. 2023

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, two novel classes of implicit exponential Runge--Kutta (ERK) methods are studied for solving highly oscillatory systems. Firstly, we analyze the symplectic conditions for two kinds of exponential integrators and obtain the symplectic method. In order to effectively solve highly oscillatory problems, we try to design the highly accurate implicit ERK integrators. By comparing the Taylor series expansion of numerical solution with exact solution, it can be verified that the order conditions of two new kinds of exponential methods are identical to classical Runge--Kutta (RK) methods, which implies that using the coefficients of RK methods, some highly accurate numerical methods are directly formulated. Furthermore, we also investigate the linear stability properties for these exponential methods. Finally, numerical results not only display the long time energy preservation of the symplectic method, but also present the accuracy and efficiency of these formulated methods in comparison with standard ERK methods.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: The Symplectic Conditions For Two New Classes Of Erk Methodsmentioning

confidence: 99%

“…1 (b) and (c), and we also plot the stability region of the explicit second-order MVERK method with two stages (see, e.g. [35]) in Fig. 1 (a).…”

Section: Linear Stabilitymentioning

confidence: 99%

“…[20,32]), Schrödinger equations (see, e.g. [3,6,35]) and KdV equations (see, e.g. [36]) can be converted into (1) with appropriate spatial discretization.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations