Implicit-explicit Runge-Kutta Methods Research Articles

Asymptotic preserving time‐discretization of optimal control problems for the Goldstein–Taylor model

We consider the development of implicit‐explicit time integration schemes for optimal control problems governed by the Goldstein–Taylor model. In the diffusive scaling, this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman–Enskog‐type limiting procedure. For the class of stiffly accurate implicit–explicit Runge–Kutta methods, the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1770–1784, 2014

A high-order discontinuous Galerkin method for fluid–structure interaction with efficient implicit–explicit time stepping

We present a high-order accurate scheme for coupled fluid–structure interaction problems. The fluid is discretized using a discontinuous Galerkin method on unstructured tetrahedral meshes, and the structure uses a high-order volumetric continuous Galerkin finite element method. Standard radial basis functions are used for the mesh deformation. The time integration is performed using a partitioned approach based on implicit–explicit Runge–Kutta methods. The resulting scheme fully decouples the implicit solution procedures for the fluid and the solid parts, which we perform using two separate efficient parallel solvers. We demonstrate up to fifth order accuracy in time on a non-trivial test problem, on which we also show that additional subiterations are not required. We solve a benchmark problem of a cantilever beam in a shedding flow, and show good agreement with other results in the literature. Finally, we solve for the flow around a thin membrane at a high angle of attack in both 2D and 3D, and compare with the results obtained with a rigid plate.

Partitioned and Implicit–Explicit General Linear Methods for Ordinary Differential Equations

Implicit–explicit (IMEX) time stepping methods can efficiently solve differential equations with both stiff and nonstiff components. IMEX Runge–Kutta methods and IMEX linear multistep methods have been studied in the literature. In this paper we study new implicit–explicit methods of general linear type. We develop an order conditions theory for high stage order partitioned general linear methods (GLMs) that share the same abscissae, and show that no additional coupling order conditions are needed. Consequently, GLMs offer an excellent framework for the construction of multi-method integration algorithms. Next, we propose a family of IMEX schemes based on diagonally-implicit multi-stage integration methods and construct practical schemes of order up to three. Numerical results confirm the theoretical findings.

High-order accurate fluid–structure simulation of a tuning fork

The aeroacoustics of a tuning fork are investigated using a high-order fluid–structure interaction (FSI) scheme. The compressible Navier–Stokes equations are discretized using a discontinuous Galerkin arbitrary Lagrangian–Eulerian (DG-ALE) method on an unstructured tetrahedral mesh, and coupled to a non-linear hyperelastic neo-Hookean model of a tuning fork, discretized using continuous Galerkin finite elements on an unstructured tetrahedral mesh. The fluid and structure are both integrated implicitly in time using a partitioned approach based on an implicit–explicit Runge–Kutta method. We measure radial sound distributions which show good agreement with theoretical predictions and physical experiments in the open literature. In addition we demonstrate how to measure Q factors for several common modes, emphasizing that we can accurately capture the decay rates arising purely from the interaction of the tuning fork with the air and without any damping built into the structure model.

Implicit–Explicit Runge–Kutta Methods for Fast–Slow Wave Problems

Abstract Linear multistage (Runge–Kutta) implicit–explicit (IMEX) time integration schemes for the time integration of fast-wave–slow-wave problems for which the fast wave has low amplitude and need not be accurately simulated are investigated. The authors focus on three-stage, second-order schemes and show that a scheme recently proposed by one of them (Kar) is unstable for purely oscillatory problems. The instability is reduced if the averaging inherent in the implicit part of the scheme is decentered, sacrificing second-order accuracy. Two alternative schemes are proposed with better stability properties for purely oscillatory problems. One of these utilizes a 3-cycle Lorenz scheme for the slow-wave terms and a trapezoidal scheme for the fast-wave terms. The other is a combination of two previously proposed schemes, which is stable for purely oscillatory problems for all fast-wave frequencies when the slow-wave frequency is less than a critical value. The alternative schemes are tested using a global spectral shallow-water model and a version of the NCEP operational global forecast model. The accuracy and stability of the alternative schemes are discussed, along with their computational efficiency.

Total-variation-diminishing implicit–explicit Runge–Kutta methods for the simulation of double-diffusive convection in astrophysics

We put forward the use of total-variation-diminishing (or more generally, strong stability preserving) implicit–explicit Runge–Kutta methods for the time integration of the equations of motion associated with the semiconvection problem in the simulation of stellar convection. The fully compressible Navier–Stokes equation, augmented by continuity and total energy equations, and an equation of state describing the relation between the thermodynamic quantities, is semi-discretized in space by essentially non-oscillatory schemes and dissipative finite difference methods. It is subsequently integrated in time by Runge–Kutta methods which are constructed such as to preserve the total variation diminishing (or strong stability) property satisfied by the spatial discretization coupled with the forward Euler method. We analyse the stability, accuracy and dissipativity of the time integrators and demonstrate that the most successful methods yield a substantial gain in computational efficiency as compared to classical explicit Runge–Kutta methods.

A discontinuous finite element baroclinic marine model on unstructured prismatic meshes

We describe the time discretization of a three-dimensional baroclinic finite element model for the hydrostatic Boussinesq equations based upon a discontinuous Galerkin finite element method. On one hand, the time marching algorithm is based on an efficient mode splitting. To ensure compatibility between the barotropic and baroclinic modes in the splitting algorithm, we introduce Lagrange multipliers in the discrete formulation. On the other hand, the use of implicit–explicit Runge–Kutta methods enables us to treat stiff linear operators implicitly, while the rest of the nonlinear dynamics is treated explicitly. By way of illustration, the time evolution of the flow over a tall isolated seamount on the sphere is simulated. The seamount height is 90% of the mean sea depth. Vortex shedding and Taylor caps are observed. The simulation compares well with results published by other authors.

A relaxation scheme for a multi-class Lighthill-Whitham-Richards traffic flow model

We present a high-resolution relaxation scheme for a multi-class Lighthill-Whitham-Richards (MCLWR) traffic flow model. This scheme is based on high-order reconstruction for spatial discretization and an implicit-explicit Runge-Kutta method for time integration. The resulting method retains the simplicity of the relaxation schemes. There is no need to involve Riemann solvers and characteristic decomposition. Even the computation of the eigenvalues is not required. This makes the scheme particularly well suited for the MCLWR model in which the analytical expressions of the eigenvalues are difficult to obtain for more than four classes of road users. The numerical results illustrate the effectiveness of the presented method.

On an accurate third order implicit-explicit Runge–Kutta method for stiff problems

Most of the popular implicit-explicit (IMEX) Runge–Kutta (R-K) methods existing in the literature suffer from the phenomenon of order reduction in the stiff regime when applied to stiff problems containing a non-stiff term and a stiff term. Specifically, order reduction is observed when the problem becomes increasingly stiff. In this paper, our motivation is to derive a third-order IMEX R-K method for stiff problems that has a better temporal order of convergence than other well-known IMEX R-K methods. A comparison with other third-order methods shows substantial potential of this new method.

On the Performance of an Implicit–Explicit Runge--Kutta Method in Models of Cardiac Electrical Activity

Mathematical models of electric activity in cardiac tissue are becoming increasingly powerful tools in the study of cardiac arrhythmias. Considered here are mathematical models based on ordinary differential equations (ODEs) that describe the ionic currents at the myocardial cell level. Generating an efficient numerical solution of these ODEs is a challenging task, and, in fact, the physiological accuracy of tissue-scale models is often limited by the efficiency of the numerical solution process. In this paper, we examine the efficiency of the numerical solution of four cardiac electrophysiological models using implicit--explicit Runge-Kutta (IMEX-RK) splitting methods. We find that variable step-size implementations of IMEX-RK methods (ARK3 and ARK5) that take advantage of Jacobian structure clearly outperform the methods commonly used in practice.

IMEX extensions of linear multistep methods with general monotonicity and boundedness properties

For solving hyperbolic systems with stiff sources or relaxation terms, time stepping methods should combine favorable monotonicity properties for shocks and steep solution gradients with good stability properties for stiff terms. In this paper we consider implicit–explicit (IMEX) multistep methods. Suitable methods will be constructed, based on explicit methods with general monotonicity and boundedness properties for hyperbolic equations. Numerical comparisons are made with several implicit–explicit Runge–Kutta methods.

Implicit–explicit numerical schemes for jump–diffusion processes

We study the numerical approximation of solutions for parabolic integro-differential equations (PIDE). Similar models arise in option pricing, to generalize the Black–Scholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Due to the non-local nature of the integral term, unconditionally stable implicit difference schemes are not practically feasible. Here we propose using implicit-explicit (IMEX) Runge-Kutta methods for the time integration to solve the integral term explicitly, giving higher-order accuracy schemes under weak stability time-step restrictions. Numerical tests are presented to show the computational efficiency of the approximation. Mathematics Subject Classification (1991): Primary: 65M12; Secondary: 35K55, 49L25

Stability of IMEX Runge–Kutta methods for delay differential equations

Stability of IMEX (implicit–explicit) Runge–Kutta methods applied to delay differential equations (DDEs) is studied on the basis of the scalar test equation d u / d t = λ u ( t ) + μ u ( t - τ ) , where τ is a constant delay and λ , μ are complex parameters. More specifically, P-stability regions of the methods are defined and analyzed in the same way as in the case of the standard Runge–Kutta methods. A new IMEX method which possesses a superior stability property for DDEs is proposed. Some numerical examples which confirm the results of our analysis are presented.

Implicit–explicit Runge–Kutta methods for financial derivatives pricing models

Implicit–explicit Runge–Kutta methods are investigated for application to financial derivatives pricing models in the partial differential equations approach. The methods are showed to be an alternative to other existing procedures for the numerical valuation of American type contracts. We follow the method of lines in order to have a numerical method that can be used with a variety of state variable discretizations including finite elements, finite differences and finite volume methods. Some numerical experiments are presented.

Pseudo-time stepping methods for space–time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

The space–time discontinuous Galerkin discretization of the compressible Navier–Stokes equations results in a non-linear system of algebraic equations, which we solve with pseudo-time stepping methods. We show that explicit Runge–Kutta methods developed for the Euler equations suffer from a severe stability constraint linked to the viscous part of the equations and propose an alternative to relieve this constraint while preserving locality. To evaluate its effectiveness, we compare with an implicit–explicit Runge–Kutta method which does not suffer from the viscous stability constraint. We analyze the stability of the methods and illustrate their performance by computing the flow around a 2D airfoil and a 3D delta wing at low and moderate Reynolds numbers.

Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation

We consider new implicit-explicit (IMEX) Runge-Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stability-preserving (SSP) scheme, and the implicit part is treated by an L-stable diagonally implicit Runge-Kutta method (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by Weighted Essentially Non Oscillatory (WENO) reconstruction. After a description of the mathematical properties of the schemes, several applications will be presented.

Implicit–explicit Runge–Kutta methods applied to atmospheric chemistry-transport modelling

Chemistry-transport calculations are highly stiff in terms of time-stepping. Because explicit ODE solvers require numerous short time steps in order to maintain stability, it seems that especially sparse implicit–explicit solvers are suited to improve the numerical efficiency for atmospheric chemistry applications. In the new version of our mesoscale chemistry-transport model MUSCAT [Knoth, O., Wolke, R., 1998a. An explicit–implicit numerical approach for atmospheric chemistry–transport modelling. Atmospheric Environment 32, 1785–1797.], implicit–explicit (IMEX) time integration schemes are implemented. Explicit second order Runge–Kutta methods for the integration of the horizontal advection are used. The stiff chemistry and all vertical transport processes (turbulent diffusion, advection, deposition) are integrated in an implicit and coupled manner utilizing the second order BDF method. The horizontal fluxes are treated as ‘artificial’ sources within the implicit integration. A change of the solution values as in conventional operator splitting is thus avoided. The aim of this paper is to investigate the interaction between the explicit Runge–Kutta scheme and the implicit integrator. The numerical behavior is discussed for a 1D test problem and 3D chemistry-transport simulations. The efficiency and accuracy of the algorithm are compared to results obtained using the Strang splitting approach. The numerical experiments indicate that our second order implicit–explicit Runge–Kutta methods are a valuable alternative to the conventional operator splitting approach for integrating atmospheric chemistry-transport-models. In mesoscale applications and in cases with stronger accuracy requirements the ‘source splitting’ approach shows a better performance than Strang splitting.

Implicit-explicit Runge-Kutta methods for computing atmospheric reactive flows

Air quality modeling is numerically extremely expensive and, therefore, it requires fast algorithms and sophisticated numerical software. This is due to the fact that reacting flow calculations are highly stiff for time-stepping. In this paper implicit-explicit time integration schemes are derived which use explicit higher order Runge-Kutta schemes for the integration of the horizontal advection. The stiff chemistry and all vertical transport processes (turbulent diffusion, advection, deposition) are integrated in an implicit and coupled manner by a higher order BDF method. High order accuracy and stability conditions are investigated for this class of implicit-explicit schemes. The numerical behavior of the new integration schemes is discussed for a 1D and a simple 3D chemistry-transport problem.

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

Implicit-explicit (IMEX) linear multistep time-discretization schemes for partial differential equations have proved useful in many applications. However, they tend to have undesirable time-step restrictions when applied to convection-diffusion problems, unless diffusion strongly dominates and an appropriate BDF-based scheme is selected (Ascher et al., 1995). In this paper, we develop Runge-Kutta-based IMEX schemes that have better stability regions than the best known IMEX multistep schemes over a wide parameter range.