Numerical study of a nonlinear heat equation for plasma physics

This paper is devoted to the numerical approximation of a nonlinear parabolic balance equation, which describes the heat evolution of a magnetically confined plasma in the edge region of a tokamak. The nonlinearity implies some numerical difficulties, in particular for the long-time behaviour approximation, when solved with standard methods. An efficient numerical scheme is presented in this paper, based on a combination of a directional splitting scheme and the implicit–explicit scheme introduced in Filbet and Jin [A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys. 229 (2010), pp. 7625–7648].     

Strong Stability Preserving Properties of Runge–Kutta Time Discretization Methods for Linear Constant Coefficient Operators

Strong stability preserving (SSP) high order Runge–Kutta time discretizations were developed for use with semi-discrete method of lines approximations of hyperbolic partial differential equations, and have proven useful in many other applications. These high order time discretization methods preserve the strong stability properties of first order explicit Euler time stepping. In this paper we analyze the SSP properties of Runge Kutta methods for the ordinary differential equation u _t =Lu where L is a linear operator. We present optimal SSP Runge–Kutta methods as well as a bound on the optimal timestep restriction. Furthermore, we extend the class of SSP Runge–Kutta methods for linear operators to include the case of time dependent boundary conditions, or a time dependent forcing term.     

On high order strong stability preserving runge-kutta and multi step time discretizations

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties-in any norm or seminorm—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge-Kutta methods for nonlinear problems and for linear problems as well as implicit Runge-Kutta methods and multi step methods will be collected.     

Assessment of a non-traditional operator split algorithm for simulation of reactive transport

A non-traditional operator split (OS) scheme for the solution of the advection-diffusion-reaction (ADR) equation is proposed. The scheme is implemented with the recently published central scheme [A. Kurganov, E. Tadmor, New high-resolution central schemes for non-linear conservation laws and convection-diffusion equations, J. Comput. Phys. 160 (2000) 241–282] to accurately simulate advection-reaction processes. The governing partial differential equation (PDE) is split into two PDEs, which are solved sequentially within each time step. Unlike traditional methods, the proposed scheme provides a very efficient method to solve the ADR equation for any value of the grid-Péclet number. An analytical mass balance error analysis shows that the proposed non-traditional scheme incurs a splitting error, which behaves differently to the splitting error incurred in traditional OS schemes. Numerical results are presented to illustrate the robustness of the proposed scheme.     

On High Order Strong Stability Preserving Runge–Kutta and Multi Step Time Discretizations

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties–in any norm or seminorm—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge–Kutta methods for nonlinear problems and for linear problems as well as implicit Runge–Kutta methods and multi step methods will be collected     

Mixed analytical/numerical method applied to the high Reynolds number k-ε turbulence model

In [Comput. Fluids 32 (2003) 659], a mixed analytical/numerical method for partial differential equation with an oscillating source term was proposed. The inhomogeneous partial differential equation is split into a homogeneous one plus an ODE for the source term using the time splitting method. The homogeneous part is then integrated numerically while the source term ODE is integrated analytically. This method was demonstrated to be efficient when the source term has a time scale much smaller than the mean flow time scale. In this paper, this approach is extended to the k-ε turbulence model for high Reynolds number flows. It is found that the mixed method based on operator splitting does not converge to a stable steady state solution. We thus propose an unsplit method. Numerical experiments for homogeneous turbulent flow and for a plane jet show that the unsplit mixed method substantially improves the accuracy of a time dependent problem while it has better convergence properties for a steady flow problem.     

ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows

We develop a new family of well-balanced path-conservative quadrature-free one-step ADER finite volume and discontinuous Galerkin finite element schemes on unstructured meshes for the solution of hyperbolic partial differential equations with non-conservative products and stiff source terms. The fully discrete formulation is derived using the recently developed framework of explicit one-step P_NP_M schemes of arbitrary high order of accuracy in space and time for conservative hyperbolic systems [Dumbser M, Balsara D, Toro EF, Munz CD. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J Comput Phys 2008;227:8209–53]. The two key ingredients of our high order approach are: first, the high order accurate P_NP_M reconstruction operator on unstructured meshes, using the WENO strategy presented in [Dumbser M, Käser M, Titarev VA Toro EF. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 2007;226:204–43] to ensure monotonicity at discontinuities, and second, a local space–time Galerkin scheme to predict the evolution of the reconstructed polynomial data inside each element during one time step to obtain a high order accurate one-step time discretization. This approach is also able to deal with stiff source terms as shown in [Dumbser M, Enaux C, Toro EF. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Comput Phys 2008;227:3971–4001]. These two key ingredients are combined with the recently developed path-conservative methods of Parés [Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21] and Castro et al. [Castro MJ, Gallardo JM, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math Comput 2006;75:1103–34] to treat the non-conservative products properly. We show applications of our method to the two-layer shallow water equations as well as applications to the recently published depth-averaged two-fluid flow model of Pitman and Le [Pitman EB, Le L. A two-fluid model for avalanche and debris flows. Philos Trans Roy Soc A 2005;363:1573–601].     

The numerical solution of the telegraph equation by the alternating group explicit (AGE) method

In this paper, the Alternating Group Explicit (AGE) method is developed from a judicious splitting of the implicit equations derived from the finite difference discretisation of the partial differential equations. The resulting equations can be reformulated in a (2?×?2) explicit form resulting in a new stable and efficient method. Finally, the AGE method is used to investigate the numerical solution of the Telegraph equation in various 2D forms.     

Numerical Study of a Modified Time-Stepping θ-Scheme for Incompressible Flow Simulations

In [Turek (1996). Int. J. Numer. Meth. Fluids 22, 987–1011], we had performed numerical comparisons for different time stepping schemes for the incompressible Navier–Stokes equations. In this paper, we present the numerical analysis in the context of the Navier–Stokes equations for a modified time-stepping θ-scheme which has been recently proposed by Glowinski [Glowinski (2003). In: Ciarlet, P. G., and Lions, J. L. (eds.), Handbook of Numerical Analysis, Vol. IX, North-Holland, Amsterdam, pp. 3–1176]. Like the well-known classical Fractional-Step-θ-scheme which had been introduced by Glowinski [Glowinski (1985). In Murman, E. M. and Abarbanel, S. S. (eds.), Progress and Supercomputing in Computational Fluid Dynamics, Birkh?user, Boston MA; Bristeau et al. (1987). Comput. Phys. Rep. 6, 73–187], too, and which is still one of the most popular time stepping schemes, with or without operator splitting techniques, this new scheme consists of 3 substeps with nonequidistant substepping to build one macro time step. However, in contrast to the Fractional-Step-θ-scheme, the second substep can be formulated as an extrapolation step for previously computed data only, and the two remaining substeps look like a Backward Euler step so that no expensive operator evaluations for the right hand side vector with older solutions, as for instance in the Crank–Nicolson scheme, have to be performed. This modified scheme is implicit, strongly A-stable and second order accurate, too, which promises some advantageous behavior, particularly in implicit CFD simulations for the nonstationary Navier–Stokes equations. Representative numerical results, based on the software package FEATFLOW [Turek (2000). FEATFLOW Finite element software for the incompressible Navier–Stokes equations: User Manual, Release 1.2, University of Dortmund] are obtained for typical flow problems with benchmark character which provide a fair rating of the solution schemes, particularly in long time simulations.Dedicated to David Gottlieb on the occasion of his 60th anniversary     

基于IMEX的织物动态仿真的近似解法

织物在空间运动的刚性特征始终是困扰织物动态仿真的难题．显式方法简单灵活，易于实现，但受稳定因素影响，无法实现具有刚性特征的织物动态模拟；隐式方法稳定性好，却忽略了非线性因素，而且计算复杂，直接影响到仿真的最终结果和实际效率．针对这一问题，提出了基于隐式一显式的近似解法，该方案从系统受力形变的非线性特征出发，将质点受力分为线性和非线性两部分，线性部分采用隐式解法，非线性部分利用显式解法，线性方程组的求解则运用近似解法．实验结果表明，该方法兼具两种方法的优点，既保留了隐式方法的稳定性，又充分利用了显式方法的简易性处理非线性特征，从而从真正意义上解决织物仿真中的刚性问题．     

On different flux splittings and flux functions in WENO schemes for balance laws

In this paper we focus our attention on obtaining well-balanced schemes for balance laws by using Marquina’s flux in combination with the finite difference and finite volume WENO schemes. We consider also the Rusanov flux splitting and the HLL approximate Riemann solver. In particular, for the presented numerical schemes we develop corresponding discretizations of the source term, based on the idea of balancing with the flux gradient. When applied to the open-channel flow and to the shallow water equations, we obtain the finite difference WENO scheme with Marquina’s flux splitting, which satisfies the approximate conservation property, and also the balanced finite volume WENO scheme with Marquina’s solver satisfying the exact conservation property. Finally, we also present an improvement of the balanced finite difference WENO scheme with the Rusanov (locally Lax–Friedrichs) flux splitting, we previously developed in [Vuković S, Sopta L. ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J Comput Phys 2002;179:593–621].     

A constitutive operator splitting method for nonlinear transient analysis

A constitutive operator splitting method for the time integration of the nonlinear dynamic equations of motion which result from the consideration of nonlinear material behavior is studied. In the method, the material constitutive law is split into a constant, history independent relation (implicit portion) and a variable history dependent relation (explicit portion); the resulting constituents are then integrated by implicit and explicit methods, respectively. Analysis of stability and some example solutions indicate that for materials with decreasing stiffness (materials which soften with increasing strain) the method is unconditionally stable. Several examples are considered which compare the accuracy of the method to exact solutions and solutions obtained by explicit central difference time integration; in all cases there is good agreement.     

Domain decomposition operator splitting for mimetic finite difference discretizations of non-stationary problems

In this paper, we propose reliable and efficient numerical methods for solving semilinear, time-dependent partial differential equations of reaction–diffusion type. The original problem is first integrated in time by using a linearly implicit fractional step Runge–Kutta method. This method takes advantage of a suitable partitioning of the diffusion operator based on domain decomposition techniques. The resulting semidiscrete problem is fully discretized by means of a mimetic finite difference method on quadrilateral meshes. Due to the previous splitting, the totally discrete scheme can be reduced to a set of uncoupled linear systems which can be solved in parallel. The overall algorithm is unconditionally stable and second-order convergent in both time and space. These properties are confirmed by numerical experiments.     

Recent Developments in the Pure Streamfunction Formulation of the Navier-Stokes System

In this paper we review fourth-order approximations of the biharmonic operator in one, two and three dimensions. In addition, we describe recent developments on second and fourth order finite difference approximations of the two dimensional Navier-Stokes equations. The schemes are compact both for the biharmonic and the Laplacian operators. For the convective term the fourth order scheme invokes also a sixth order Pade approximation for the first order derivatives, using an approximation suggested by Carpenter-Gottlieb-Abarbanel (J. Comput. Phys. 108:272–295, 1993). We also introduce the derivation of a pure streamfunction formulation for the Navier-Stokes equations in three dimensions.     

The positivity of low-order explicit Runge-Kutta schemes applied in splitting methods

Splitting methods are frequently used for the solution of large stiff initial value problems of ordinary differential equations with an additively split right-hand side function. Such systems arise, for instance, as method of lines discretizations of evolutionary partial differential equations in many applications. We consider the choice of explicit Runge-Kutta (RK) schemes in implicit-explicit splitting methods. Our main objective is the preservation of positivity in the numerical solution of linear and nonlinear positive problems while maintaining a sufficient degree of accuracy and computational efficiency. A three-stage second-order explicit RK method is proposed which has optimized positivity properties. This method compares well with standard s-stage explicit RK schemes of order s, s = 2, 3. It has advantages in the low accuracy range, and this range is interesting for an application in splitting methods. Numerical results are presented.     

Pricing American options under jump-diffusion models using local weak form meshless techniques

Recently, several numerical methods have been proposed for pricing options under jump-diffusion models but very few studies have been conducted using meshless methods [R. Chan and S. Hubbert, A numerical study of radial basis function based methods for options pricing under the one dimension jump-diffusion model, Tech. Rep., 2010; A. Saib, D. Tangman, and M. Bhuruth, A new radial basis functions method for pricing American options under Merton's jump-diffusion model, Int. J. Comput. Math. 89 (2012), pp. 1164–1185]. Indeed, only a strong form of meshless methods have been employed in these lectures. We propose the local weak form meshless methods for option pricing under Merton and Kou jump-diffusion models. Predominantly in this work we will focus on meshless local Petrov–Galerkin, local boundary integral equation methods based on moving least square approximation and local radial point interpolation based on Wendland's compactly supported radial basis functions. The key feature of this paper is applying a Richardson extrapolation technique on American option which is a free boundary problem to obtain a fixed boundary problem. Also the implicit–explicit time stepping scheme is employed for the time derivative which allows us to obtain a spars and banded linear system of equations. Numerical experiments are presented showing that the presented approaches are extremely accurate and fast.     

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.     

Higher order operator splitting methods via Zassenhaus product formula: Theory and applications

In this paper, we contribute higher order operator splitting methods improved by Zassenhaus product. We apply the contribution to classical and iterative splitting methods. The underlying analysis to obtain higher order operator splitting methods is presented. While applying the methods to partial differential equations, the benefits of balancing time and spatial scales are discussed to accelerate the methods.The verification of the improved splitting methods are done with numerical examples. An individual handling of each operator with adapted standard higher order time-integrators is discussed. Finally, we conclude the higher order operator splitting methods.     

Fast methods for the iterative solution of linear elliptic and parabolic partial differential equations involving 2 space dimensions

Within the last decade, attention has been devoted to the introduction of several fast computational methods for solving the linear difference equations which are derived from the finite difference discretisation of many standard partial differential equations of Mathematical Physics.

In this paper, the authors develop and extend an exact factorisation technique previously applied to parabolic equations in one space dimension to the implicit difference equations which are derived from the application of alternating direction implicit methods when applied to elliptic and parabolic partial differential equations in 2 space dimensions under a variety of boundary conditions.     

A New Mapped Weighted Essentially Non-oscillatory Scheme

The weighted essentially non-oscillatory (WENO) methods are a popular high-order spatial discretization for hyperbolic partial differential equations. Recently Henrick et al. (J. Comput. Phys. 207:542–567, 2005) noted that the fifth-order WENO method by Jiang and Shu (J. Comput. Phys. 126:202–228, 1996) is only third-order accurate near critical points of the smooth regions in general. Using a simple mapping function to the original weights in Jiang and Shu (J. Comput. Phys. 126:202–228, 1996), Henrick et al. developed a mapped WENO method to achieve the optimal order of accuracy near critical points. In this paper we study the mapped WENO scheme and find that, when it is used for solving the problems with discontinuities, the mapping function in Henrick et al. (J. Comput. Phys. 207:542–567, 2005) may amplify the effect from the non-smooth stencils and thus cause a potential loss of accuracy near discontinuities. This effect may be difficult to be observed for the fifth-order WENO method unless a long time simulation is desired. However, if the mapping function is applied to seventh-order WENO methods (Balsara and Shu in J. Comput. Phys. 160:405–452, 2000), the error can increase much faster so that it can be observed with a moderate output time. In this paper a new mapping function is proposed to overcome this potential loss of accuracy.