Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations

Ascher, Uri M.; Ruuth, Steven J.; Spiteri, Raymond J.

doi:10.1016/s0168-9274(97)00056-1

Cited by 996 publications

(1,027 citation statements)

References 11 publications

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“…Since theL-operator is diagonal, the need to invert theL term does not incur any extra computational cost. This idea has appeared many times in the literature for both multistep and Runge-Kutta methods, often for diffusive problems and bearing a name such as "implicit-explicit" or "linearly implicit" [1][2][3][4][5]. Graphically, we illustrate a scheme of the form (4) in Fig.…”

Section: Linearly Implicit Multistep Methods For Odesmentioning

confidence: 99%

See 1 more Smart Citation

A Fast Spectral Algorithm for Nonlinear Wave Equations with Linear Dispersion

Fornberg¹,

Driscoll²

1999

Journal of Computational Physics

115

104

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Section: Linearly Implicit Multistep Methods For Odesmentioning

confidence: 99%

“…One gets second-order accuracy in time by alternating A, B, A in the equations above while using the time increments 1 4 , 1 2 , 1 4 . Yoshida [18] showed a systematic way to find split-step methods of any even order.…”

Section: Ssmentioning

confidence: 99%

A Fast Spectral Algorithm for Nonlinear Wave Equations with Linear Dispersion

Fornberg¹,

Driscoll²

1999

Journal of Computational Physics

115

104

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“…The density is a constant and p=RT. The numerical performance of the proposed methods were compared with other reported ARK2 methods in the literature, including the (2,3,3), (3,4,3), and (4,4,3) methods of Ascher et al [30], LIRK3 and LIRK4 due to Calvo et al [31], a five-stage, 3(2) pair of Fritzen and Wittekindt (FW53) [32]. Figure 5 show some numerical results of pressure, temperature, temperature gradient and mole fraction of major species and minor species.…”

Section: Constant Volume Explosion Modelmentioning

confidence: 99%

Additive Runge-Kutta methods for H2/O2/Ar detonation with a detailed elementary chemical reaction model

Ren

Ning

2013

Chin. Sci. Bull.

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We report here the additive Runge-Kutta methods for computing reactive Euler equations with a stiff source term, and in particular, their applications in gaseous detonation simulations. The source term in gaseous detonation is stiff due to the presence of wide range of time scales during thermal-chemical non-equilibrium reactive processes and some of these time scales are much smaller than that of hydrodynamic flow. The high order, L-stable, additive Runge-Kutta methods proposed in this paper resolved the stiff source term into the stiff part and non-stiff part, in which the stiff part was solved implicitly while the non-stiff part was handled explicitly. The proposed method was successfully applied to simulating the gaseous detonation in a stoichiometric H 2 /O 2 /Ar mixture based on a detailed elementary chemical reaction model comprised of 9 species and 19 elementary reactions. The results showed that the stiffly accurate additive Runge-Kutta methods can capture the discontinuity well, and describe the detonation complex wave configurations accurately such as the triple wave structure and cellular pattern. Reacting flows, specifically in gaseous combustion, have been a significant topic of active research for more than one hundred years. The strong coupling between hydrodynamic flow and chemical kinetics is complex and even today many phenomena are not very well understood yet. Gaseous detonation is a process of supersonic combustion in which a shock wave is propagated and supported by the energy release in a reaction zone behind it. It is the more powerful and destructive of the two general classes of combustion, the other one being deflagration. The primary difficulty in computing reacting flows is the source term stiffness inherent in the reactive Euler equations in temporal integrations. Besides, the viscous stress and heat flux terms in the boundary layers can cause the stiffness too. The source terms are stiff because the thermal-chemical non-equilibrium reactive processes possess a wide range of time scales and some of them are much smaller than that of hydrodynamic flow [1]. The simulation will be inefficient when the explicit methods rather than the implicit methods are used, because the time-step sizes dictated by the stability restraint in explicit methods are much smaller than those required by the CFL condition. Due to these limitations in explicit methods, the implicit methods are normally required to simulate gaseous detonation. The practical implicit methods for gaseous detonation simulation can be categorized into two classes, i.e. the time-splitting method and the additive semi-implicit method [2,3].The time-splitting methods [4][5][6][7][8], resolve the source term of reactive Euler equations into ( ) ( ),   t

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“…A high order time semidiscretization of (3.1) can be achieved by using an IMEX scheme as described in [5,39], which allows us to treat implicitly the stiff term on the right-hand side and to keep explicit the linear left-hand side of (3.1). Moreover, since the adopted IMEX scheme is only diagonally implicit, each implicit equation can be solved autonomously and does not match with the other equations.…”

Section: Time Semidiscretizationmentioning

confidence: 99%

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

Cavalli¹,

Naldi²,

Perugia³

2012

SIAM J. Sci. Comput.

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Abstract. In this work we present finite element approximations of relaxed systems for nonlinear diffusion problems, which can also tackle the cases of degenerate and strongly degenerate diffusion equations. Relaxation schemes take advantage of the replacement of the original partial differential equation (PDE) with a semilinear hyperbolic system of equations, with a stiff source term, tuned by a relaxation parameter ε. When ε → 0 + , the system relaxes onto the original PDE: in this way, a consistent discretization of the relaxation system for vanishing ε yields a consistent discretization of the original PDE. The numerical schemes obtained with this procedure do not require solving implicit nonlinear problems and possess the robustness of upwind discretizations. The proposed approximations are based on a discontinuous Galerkin method in space and on suitable implicitexplicit integration in time. Then, in principle, we can achieve any order of accuracy and obtain stable solutions, even when the diffusion equation becomes degenerate and solution singularities develop. Moreover, when needed, we can easily incorporate slope limiters within our schemes in order to handle spurious oscillatory phenomena. Some preliminary theoretical results are given, along with several numerical tests in one and two space dimensions, both for linear and nonlinear diffusion problems, including a degenerate diffusion equation, that provide numerical evidence of the properties of the presented approach. Key words. discontinuous Galerkin method, relaxation models, nonlinear diffusion AMS subject classifications. 65M60, 65M12, 35K55DOI. 10.1137/110827752 1. Introduction. Linear and nonlinear diffusion equations come from a variety of diffusion phenomena widely appearing in nature. They are suggested as mathematical models in many fields, such as filtration, phase transition, biochemistry, image analysis, and dynamics of biological groups. In the nonlinear case, the classical solutions to many of these PDEs fail to exist in finite time, even if the initial data are smooth. In such cases, suitable criteria have been introduced, which allow one to select physically relevant weak solutions beyond the singularity time.Recently, relaxation approximations to such PDEs have been introduced. These methods are based on replacing the equation by a semilinear hyperbolic system with stiff relaxation terms, tuned by a relaxation parameter ε. When ε → 0 + , the solution of this system "relaxes" onto the solution of the original PDE. Thus a consistent discretization of the relaxation system for ε = 0 yields a consistent discretization of the original PDE, as can be seen, for instance, in [29] and [2]. The advantage of this procedure is that the numerical scheme obtained in this fashion does not need approximate Riemann solvers for the convective term but possesses the robustness of upwind discretizations. Moreover, the complexity introduced by replacing the original

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Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations

Cited by 996 publications

References 11 publications

A Fast Spectral Algorithm for Nonlinear Wave Equations with Linear Dispersion

A Fast Spectral Algorithm for Nonlinear Wave Equations with Linear Dispersion

Additive Runge-Kutta methods for H2/O2/Ar detonation with a detailed elementary chemical reaction model

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

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