On the Linear Stability of the Fifth-Order WENO Discretization

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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“…The latter requirement is associated with a stable integration of the hyperbolic advection terms (see [24,36]). …”

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confidence: 99%

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

et al. 2012

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“…In fact, these authors found out both in terms of linear stability theory and in simple numerical examples that forward Euler does not do well when complementing the WENO5 semidiscretization. Thus, the method that SSP methods "want to be like" is nothing to aspire to in the WENO context; see also [2,28].…”

Section: Ssp Methodsmentioning

confidence: 99%

“…But then WENO is unnecessarily working on the imperfection of the time discretization scheme. The net effect is the necessity when working with forward Euler of occasionally taking much smaller time steps than would otherwise be allowed; see also [28]. So, the SSP concept in its unmodified form may be irrelevant when a WENO semi-discretization is employed.…”

Section: Ssp Methodsmentioning

confidence: 99%

Surprising computations

Ascher

2012

Applied Numerical Mathematics

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In the course of simulation of differential equations, especially of marginally stable differential problems using marginally stable numerical methods, one occasionally comes across a correct computation that yields surprising, or unexpected results. We examine several instances of such computations. These include (i) solving Hamiltonian ODE systems using almost conservative explicit Runge-Kutta methods, (ii) applying splitting methods for the nonlinear Schrödinger equation, and (iii) applying strong stability preserving Runge-Kutta methods in conjunction with weighted essentially non-oscillatory semi-discretizations for nonlinear conservation laws with discontinuous solutions.For each problem and method class we present a simple numerical example that yields results that in our experience many active researchers are finding unexpected and unintuitive. Each numerical example is then followed by an explanation and a resolution of the practical problem.

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“…According to Wang and Spiteri [13], they are all linearly unstable in theory when coupled with the WENO5 scheme except TVD3. But the Courant numbers we use are small enough in terms of Motamed et al [20] to make the combination with WENO5 stable in practical applications. …”

Section: Definition 2 Assume Thatmentioning

confidence: 99%

“…The results are given for the smooth initial condition (19) in Tables A.14, A.15, A.16 and A.17 for the Euler forward, the TVD2, the TVD3, and the SSP RK(3,2) scheme, respectively. For the discontinuous initial condition (20), they can be found in Tables B.18, B.19, B.20 and B.21. In each row, the spatial resolution is fixed, whereas in the columns, the temporal resolution is constant.…”

Section: Errors Of Runge-kutta Schemesmentioning

confidence: 99%

Achievable efficiency of numerical methods for simulations of solar surface convection

Grimm‐Strele

Kupka

Muthsam

2015

Computer Physics Communications

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We investigate the achievable efficiency of both the time and the space discretisation methods used in Antares for mixed parabolichyperbolic problems. We show that the fifth order variant of WENO combined with a second order Runge-Kutta scheme is not only more accurate than standard first and second order schemes, but also more efficient taking the computation time into account. Then, we calculate the error decay rates of WENO with several explicit Runge-Kutta schemes for advective and diffusive problems with smooth and non-smooth initial conditions. With this data, we estimate the computational costs of three-dimensional simulations of stellar surface convection and show that SSP RK(3,2) is the most efficient scheme considered in this comparison.Keywords: Methods: numerical, Numerical astrophysics, Runge-Kutta schemes, efficiency, WENO scheme, HydrodynamicsThe simulation code Antares [1] was developed for the simulation of solar and stellar surface convection. Recently it has also been applied to many other astrophysical problems [e.g. 2, 3].In this code, the Navier-Stokes equations (usually without magnetic field) and with radiative transfer (radiation hydrodynamics, RHD) are solved in the form∂E ∂t + ∇ · (u (E + p)) = ρ (g · u) + ∇ · (u · τ) + Q rad .

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On the Linear Stability of the Fifth-Order WENO Discretization

Cited by 26 publications

References 21 publications

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

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

Surprising computations

Achievable efficiency of numerical methods for simulations of solar surface convection

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