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

Christlieb, Andrew; Gottlieb, Sigal; Grant, Zachary J.; Seal, David C.

doi:10.1007/s10915-016-0164-2

Cited by 49 publications

(63 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We shift (x j , t n ) to (0, 0) due to the invariance of (16) with respect to the translation of coordinates. In order to proceed in one of those frameworks, we have to solve (16) approximately subject to the initial data…”

Section: Lax-wendroff Flow Solvers For Nonlinear Hyperbolic Balance Lawsmentioning

confidence: 99%

Two-stage fourth order: temporal-spatial coupling in computational fluid dynamics (CFD)

2019

Adv. Aerodyn.

View full text Add to dashboard Cite

With increasing engineering demands, there need high order accurate schemes embedded with precise physical information in order to capture delicate small scale structures and strong waves with correct"physics". There are two families of high order methods: One is the method of line, relying on the Runge-Kutta (R-K) time-stepping. The building block is the Riemann solution labeled as the solution element "1". Each step in R-K just has first order accuracy. In order to derive a fourth order accuracy scheme in time, one needs four stages labeled as "1 ⊙ 1 ⊙ 1 ⊙ 1 = 4". The other is the one-stage Lax-Wendroff (LW) type method, which is more compact but is complicated to design numerical fluxes and hard to use when applied to highly nonlinear problems. In recent years, the pair of solution element and dynamics element, labeled as "2", are taken as the building block. The direct adoption of the dynamics implies the inherent temporal-spatial coupling. With this type of building blocks, a family of two-stage fourth order accurate schemes, labeled as "2⊙2 = 4", are designed for the computation of compressible fluid flows. The resulting schemes are compact, robust and efficient. This paper contributes to elucidate how and why high order accurate schemes should be so designed. To some extent, the "2 ⊙ 2 = 4" algorithm extracts the advantages of the method of line and one-stage LW method. As a core part, the pair "2" is expounded and LW solver is revisited. The generalized Riemann problem (GRP) solver, as the discontinuous and nonlinear version of LW flow solver, and the gas kinetic scheme (GKS) solver, the microscopic LW solver, are all reviewed. The compact Hermite-type data reconstruction and high order approximation of 2 Two-stage fourth order: Temporal Spatial Coupling in CFD boundary conditions are proposed. Besides, the computational performance and prospective discussions are presented.Keywords Compressible fluid dynamics · hyperbolic balance laws · high order methods · temporal-spatial coupling · multi-stage two-derivative methods · Lax-Wendroff type flow solvers · GRP solver

show abstract

Section: Lax-wendroff Flow Solvers For Nonlinear Hyperbolic Balance Lawsmentioning

confidence: 99%

Two-stage fourth order: temporal-spatial coupling in computational fluid dynamics (CFD)

2019

Adv. Aerodyn.

View full text Add to dashboard Cite

show abstract

“…If we consider the effective observed time-step (i.e. the observed time-step normalized by the number of stages s) we see that the SSPIFRK (9,4) has C ef f obs = 0.47 while the SSPIF-TSRK (3,4) has a smaller C ef f obs = 0.42 and SSPIF-TSRK(4, 4) has an even smaller C ef f obs = 0.4. However, we observe that the allowable TVD time-step of the SSPIFRK methods with (s, p) = (5, 4), (9,4) is smaller than that of the corresponding SSPIF-TSRK methods.…”

Section: Examplementioning

confidence: 99%

“…We consider a selection of fourth order methods. First, we explore the performance of the SSPIF-TSRK(s, p) methods with (s, p) = (3,4), (4,4) (the corresponding SSPIFRK methods do not exist). If we consider the effective observed time-step (i.e.…”

Section: Examplementioning

confidence: 99%

“…They have since been developed extensively and have been widely used to preserve different numerical stability properties needed in a variety of application areas. Furthermore, many classes of SSP time-stepping methods have been studied, including SSP explicit and implicit linear multi-step methods [9], Runge-Kutta methods [29,30,18,19], multi-stage multi-step methods [2,22,23], and multi-stage multi-derivative methods [3,12].…”

mentioning

confidence: 99%

See 1 more Smart Citation

Strong Stability Preserving Integrating Factor Two-Step Runge–Kutta Methods

Isherwood

Grant

2019

J Sci Comput

Self Cite

View full text Add to dashboard Cite

Problems with components that feature significantly different time scales, where the stiff time-step restriction comes from a linear component, implicit-explicit (IMEX) methods alleviate this restriction if the concern is linear stability. However, when nonlinear non-inner-product stability properties are of interest, such as in the evolution of hyperbolic partial differential equations with shocks or sharp gradients, linear inner-product stability is no longer sufficient for convergence, and so strong stability preserving (SSP) methods are often needed. Where the SSP property is needed, IMEX SSP Runge-Kutta (SSP-IMEX) methods have very restrictive time-steps. An alternative to SSP-IMEX schemes is to adopt an integrating factor approach to handle the linear component exactly and step the transformed problem forward using some time-evolution method. The strong stability properties of integrating factor Runge-Kutta methods were established in [15], where it was shown that it is possible to define explicit integrating factor Runge-Kutta methods that preserve strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping. It was proved that the solution will be SSP if the transformed problem is stepped forward with an explicit SSP Runge-Kutta method that has non-decreasing abscissas. However, explicit SSP Runge-Kutta methods have an order barrier of p = 4, and sometimes higher order is desired. In this work we consider explicit SSP two-step Runge-Kutta integrating factor methods to raise the order. We show that strong stability is ensured if the two-step Runge-Kutta method used to evolve the transformed problem is SSP and has non-decreasing abscissas. We find such methods up to eighth order and present their SSP coefficients. Adding a step allows us to break the fourth order barrier on explicit SSP Runge-Kutta methods; furthermore, our explicit SSP two-step Runge-Kutta methods with non-decreasing abscissas typically have larger SSP coefficients than the corresponding one-step methods. A selection of our methods are tested for convergence and demonstrate the design order. We also show, for selected methods, that the SSP time-step predicted by the theory is a lower bound of the allowable time-step for linear and nonlinear problems that satisfy the total variation diminishing (TVD) condition. We compare some of the non-decreasing abscissa SSP two-step Runge-Kutta methods to previously found methods that do not satisfy this criterion on linear and nonlinear TVD test cases to show that this non-decreasing abscissa condition is indeed necessary in practice as well as theory. We also compare these results to our SSP integrating factor Runge-Kutta methods designed in [15]. This paper is dedicated to the memory of Saul Abarbanel. His wisdom, humor, and kindness were a gift to all who knew him.

show abstract

“…An outstanding method is the two-stage fourth-order scheme for the Euler equations [27], where both the flux and its time derivative are used in the construction of higher-order scheme. The two-stage fourth-order algorithms have been developed under the multi-stage multi-derivative (MSMD) framework [15,42,9]. Similar to the GRP method, the gas-kinetic scheme (GKS) is also based on a time accurate flux function at a cell interface [54,53,55].…”

Section: Introductionmentioning

confidence: 99%

Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations

Zhao

Shyy

et al. 2019

Adv. Aerodyn.

View full text Add to dashboard Cite

In this paper, a class of compact higher-order gas-kinetic schemes (GKS) with spectral resolution will be presented. Based on the high-order gas evolution model in GKS, both the interface flux function and conservative flow variables can be evaluated explicitly from the time-accurate gas distribution function. As a result, inside each control volume both the cell-averaged flow variables and their cell-averaged gradients can be updated within each time step. The flow variable update and slope update are coming from the same physical solution at the cell interface. Different from many other approaches, such as HWENO, there are no additional governing equations in GKS for the slopes or equivalent degrees of freedom independently inside each cell. Therefore, based on both cell averaged values and their slopes, compact 6th-order and 8th-order linear and nonlinear reconstructions can be developed. As analyzed in this paper, the local linear compact reconstruction can achieve a spectral-like resolution at large wavenumber than the well-established compact scheme of Lele with globally coupled flow variables and their derivatives. For nonlinear gas dynamic evolution, in order to avoid spurious oscillation in discontinuous region, the above compact linear reconstruction from the symmetric stencil can be divided into sub-stencils and apply a biased nonlinear WENO-Z reconstruction. Consequently discontinuous solutions can be captured through the 6th-order and 8th-order compact WENO-type nonlinear reconstruction. In GKS, the time evolution solution of the gas distribution function at a cell interface is based on an integral solution of the kinetic model equation, which covers a physical process from an initial non-equilibrium state to a final equilibrium one. Since the initial non-equilibrium state is obtained based on the nonlinear WENO-Z reconstruction, and the equilibrium state is basically constructed from the linear symmetric reconstruction, the GKS evolution models unifies the nonlinear and linear reconstructions in gas evolution process for the determination of a time-dependent gas distribution function, which gives great advantages in capturing both discontinuous shock wave and the linear aero-acoustic wave in a single computation due to dynamical adaptation of non-equilibrium and equilibrium state from GKS evolution model in different regions. This dynamically adaptive model helps to solve a long lasting problem in the development of high-order schemes about the choices of the linear and nonlinear reconstructions. Compared with discontinuous Galerkin 1 arXiv:1901.00261v1 [physics.comp-ph] 2 Jan 2019 (DG) scheme, the current compact GKS uses the same local and compact stencil, achieves the 6th-order and 8th-order accuracy, uses a much larger time step with CFL number ≥ 0.3, has the robustness as a 2nd-order scheme, and gets accurate solutions in both shock and smooth regions without introducing trouble cell and additional limiting process. The nonlinear reconstruction in the compact scheme is solely based on the WENO-...

show abstract

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

Cited by 49 publications

References 36 publications

Two-stage fourth order: temporal-spatial coupling in computational fluid dynamics (CFD)

Two-stage fourth order: temporal-spatial coupling in computational fluid dynamics (CFD)

Strong Stability Preserving Integrating Factor Two-Step Runge–Kutta Methods

Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations

Contact Info

Product

Resources

About