Two-derivative Runge–Kutta methods for PDEs using a novel discretization approach

Tsai, Angela; Chan, R. P. K.; Wang, Shixiao

doi:10.1007/s11075-014-9823-2

Cited by 39 publications

(14 citation statements)

References 7 publications

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“…It is also possible to combine these approaches to obtain methods with multiple steps, stages, and derivatives. Multistage multiderivative integration methods were first considered in [24,35,38], and multiderivative time integrators for ordinary differential equations have been developed in [3,14,15,22,25,30,31], but only recently have these methods been explored for use with partial differential equations (PDEs) [29,37]. In this work, we consider explicit multistage two-derivative time integrators as applied to the numerical solution of hyperbolic conservation laws.…”

Section: Multistage Multiderivative Methodsmentioning

confidence: 99%

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Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes

et al. 2016

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High order strong stability preserving (SSP) time discretizations are advantageous for use with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. The search for high order strong stability time-stepping methods with large allowable strong stability time-step has been an active area of research over the last two decades. Recently, multiderivative time-stepping methods have been implemented with hyperbolic PDEs. In this work we describe sufficient conditions for a two-derivative multistage method to be SSP, and find some optimal SSP multistage two-derivative methods. While explicit SSP Runge-Kutta methods exist only up to fourth order, we show that this order barrier is broken for explicit multi-stage two-derivative methods by designing a three stage fifth order SSP method. These methods are tested on simple scalar PDEs to verify the order of convergence, and demonstrate the need for the SSP condition and the sharpness of the SSP time-step in many cases.

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Section: Multistage Multiderivative Methodsmentioning

confidence: 99%

“…Recently, multi-stage multiderivative methods have been proposed for use with hyperbolic PDEs [29,37]. The question then arises as to whether these methods can be strong stability preserving as well.…”

mentioning

confidence: 99%

Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes

et al. 2016

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“…The explicit two-stage fourth-order accurate time discretizations are studied in [5,7] and successfully applied to the nonlinear hyperbolic conservation laws. They belong to the two-derivative Runge-Kutta methods, see [3,1,6]. In comparison with the explicit four-stage fourth-order accurate Runge-Kutta method, they only calls the time-consuming exact or approximate Riemann solver and the initial reconstruction with the characteristic decomposition twice at each time step, which is half of the former.…”

Section: Introductionmentioning

confidence: 99%

On the explicit two-stage fourth-order accurate time discretizations

Yuan¹,

Tang²

2020

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This paper continues to study the explicit two-stage fourth-order accurate time discretizations [5,7]. By introducing variable weights, we propose a class of more general explicit one-step two-stage time discretizations, which are different from the existing methods, such as the Euler methods, Runge-Kutta methods, and multistage multiderivative methods etc.We study the absolute stability, the stability interval, and the intersection between the imaginary axis and the absolute stability region. Our results show that our two-stage time discretizations can be fourth-order accurate conditionally, the absolute stability region of the proposed methods with some special choices of the variable weights can be larger than that of the classical explicit fourth-or fifth-order Runge-Kutta method, and the interval of absolute stability can be almost twice as much as the latter. Several numerical experiments are carried out to demonstrate the performance and accuracy as well as the stability of our proposed methods.

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“…Although seldom used, the multistage multiderivative methods have been investigated (for ODEs) since as early as the 1960's for problems in celestial mechanics [43], and later on for various other differential equations [20,19]. The multistage multiderivative flavor of these solvers have only recently attracted attention as a mechanism for discretizing partial differential equations [49,45]. In [13] it is shown that the multistage multiderivative formulations can be constructed to contain the so-called strong stability preserving property, and these solvers are currently being investigated as useful time discretizations for equations of gas dynamics [34,39,38].…”

Section: Introductionmentioning

confidence: 99%

Implicit Multiderivative Collocation Solvers for Linear Partial Differential Equations with Discontinuous Galerkin Spatial Discretizations

2017

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In this work, we construct novel discretizations for the unsteady convectiondiffusion equation. Our discretization relies on multiderivative time integrators together with a novel discretization that reduces the total number of unkowns for the solver. These type of temporal discretizations come from an umbrella class of methods that include Lax-Wendroff (Taylor) as well as Runge-Kutta methods as special cases. We include two-point collocation methods with multiple time derivatives as well as a sixth-order fully implicit collocation method that only requires a total of three stages. Numerical results for a number of sample linear problems indicate the expected order of accuracy and indicate we can take arbitrarily large time steps.

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Two-derivative Runge–Kutta methods for PDEs using a novel discretization approach

Cited by 39 publications

References 7 publications

Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes

Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes

On the explicit two-stage fourth-order accurate time discretizations

Implicit Multiderivative Collocation Solvers for Linear Partial Differential Equations with Discontinuous Galerkin Spatial Discretizations

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