Optimal implicit strong stability preserving Runge–Kutta methods

Ketcheson, David I.; Macdonald, Colin B.; Gottlieb, Sigal

doi:10.1016/j.apnum.2008.03.034

Cited by 99 publications

(122 citation statements)

References 25 publications

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“…We do not consider k = 0 because, as we have seen in the previous subsection (see (3.19)), for the values of γ satisfying (3.20), it holds that R(z) ≥ 0 for z ≤ 0. Furthermore, we restrict the values to −x ∈ [0, 6] because, by numerical search, it has been found that the nonlinear optimum radius of absolute monotonicity for second order 3-stage SDIRK schemes is 6 [4,19]. Consequently, for −x ∈ [0, 6], K(x) ≤ 4.…”

Section: Radius Of Absolute Monotonicity For Linear Problemsmentioning

confidence: 99%

“…This is a second order IMEX method such that b =b. The explicit part is the optimal three-stage explicit SSP method of second order [21], while the implicit part represents the optimal three-stage scheme [3,4,19]. The stability function for the explicit scheme is given by (4.3), while for the implicit scheme it is (see, e.g., [20])…”

Section: Imex Scheme With Large Region Of Absolute Monotonicitymentioning

confidence: 99%

“…For RK and IMEX RK methods, step size restrictions to obtain SSP or TVD schemes are given, respectively, by the Kraaijevanger radius (or radius of absolute monotonicity) and the region of absolute monotonicity. The literature collects an extensive research on TVD and SSP RK methods [3,4,10,11,12,13,14,17,18,21,22,26,27,28,30,31,32,33] (see [5,7,9,29] for reviews on the topic).…”

Section: It Turns Out That Ifmentioning

confidence: 99%

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Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity

et al. 2012

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Abstract. We construct and analyze strong stability preserving implicit-explicit Runge-Kutta methods for the time integration of models of flow and radiative transport in astrophysical applications. It turns out that in addition to the optimization of the region of absolute monotonicity, other properties of the methods are crucial for the success of the simulations as well. The models in our focus dictate to also take into account the step-size limits associated with dissipativity and positivity of the stiff parabolic terms which represent transport by diffusion. Another important property is uniform convergence of the numerical approximation with respect to different stiffness of those same terms. Furthermore, it has been argued that in the presence of hyperbolic terms, the stability region should contain a large part of the imaginary axis even though the essentially non-oscillatory methods used for the spatial discretization have eigenvalues with a negative real part. Hence, we construct several new methods which differ from each other by taking various or even all of these constraints simultaneously into account. It is demonstrated for the problem of double-diffusive convection that the newly constructed schemes provide a significant computational advantage over methods from the literature. They may also be useful for general problems which involve the solution of advectiondiffusion equations, or other transport equations with similar stability requirements.

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Section: Radius Of Absolute Monotonicity For Linear Problemsmentioning

confidence: 99%

Section: Imex Scheme With Large Region Of Absolute Monotonicitymentioning

confidence: 99%

Section: It Turns Out That Ifmentioning

confidence: 99%

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Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity

et al. 2012

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“…These spatial discretizations are often combined with strong stability preserving (SSP) Runge-Kutta and multistep timestepping schemes. SSP time-stepping schemes preserve convex boundedness and contractivity properties of the spatial discretization combined with forward Euler, under a modified time-step restriction [20,18,3,4,16,11].…”

Section: Introductionmentioning

confidence: 99%

On the Linear Stability of the Fifth-Order WENO Discretization

2010

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We study the linear stability of the fifth-order Weighted Essentially Non-Oscillatory spatial discretization (WENO5) combined with explicit time stepping applied to the one-dimensional advection equation. We show that it is not necessary for the stability domain of the time integrator to include a part of the imaginary axis. In particular, we show that the combination of WENO5 with either the forward Euler method or a two-stage, second-order Runge-Kutta method is linearly stable provided very small time step-sizes are taken. We also consider fifth-order multistep time discretizations whose stability domains do not include the imaginary axis. These are found to be linearly stable with moderate time steps when combined with WENO5. In particular, the fifth-order extrapolated BDF scheme gave superior results in practice to high-order Runge-Kutta methods whose stability domain includes the imaginary axis. Numerical tests are presented which confirm the analysis.

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“…Note that this is only a choice, and that other time-stepping schemes are also possible [13,21,33]. where s 0 0 are the scaling-function coefficients belonging to u h ; the multiwavelets on higher levels are defined as ψ m j (x) = 2 m/2 ψ (2 m (x+1)− 2j − 1); and d m j are the corresponding multiwavelet coefficients [3,39], which are determined using the orthogonal projection of the DG approximation onto the multiwavelet basis:…”

mentioning

confidence: 99%

Automated Parameters for Troubled-Cell Indicators Using Outlier Detection

Vuik

Ryan

2016

SIAM J. Sci. Comput.

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Abstract. In Vuik and Ryan [J. Comput. Phys., 270 (2014), pp. 138-160] we studied the use of troubled-cell indicators for discontinuity detection in nonlinear hyperbolic partial differential equations and introduced a new multiwavelet technique to detect troubled cells. We found that these methods perform well as long as a suitable, problem-dependent parameter is chosen. This parameter is used in a threshold which decides whether or not to detect an element as a troubled cell. Until now, these parameters could not be chosen automatically. The choice of the parameter has an impact on the approximation: it determines the strictness of the troubled-cell indicator. An inappropriate choice of the parameter will result in the detection (and limiting) of too few or too many elements. The optimal parameter is chosen such that the minimal number of troubled cells is detected and the resulting approximation is free of spurious oscillations. In this paper we will see that for each troubled-cell indicator the sudden increase or decrease of the indicator value with respect to the neighboring values is important for detection. Indication basically reduces to detecting the outliers of a vector (one dimension) or matrix (two dimensions). This is done using Tukey's boxplot approach to detect which coefficients in a vector are straying far beyond the others [J. W. Tukey, Exploratory Data Analysis, Addison-Wesley, 1977]. We provide an algorithm that can be applied to various troubled-cell indication variables. With this technique, the problem-dependent parameter that the original indicator requires is no longer necessary as the parameter will be chosen automatically.

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Optimal implicit strong stability preserving Runge–Kutta methods

Cited by 99 publications

References 25 publications

Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity

Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity

On the Linear Stability of the Fifth-Order WENO Discretization

Automated Parameters for Troubled-Cell Indicators Using Outlier Detection

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