Runge Kutta Chebyshev Research Articles

An Eulerian–Lagrangian method for coupled parabolic-hyperbolic equations

We present an Eulerian–Lagrangian method for the numerical solution of coupled parabolic-hyperbolic equations. The method combines advantages of the modified method of characteristics to accurately solve the hyperbolic equations with an Eulerian method to discretize the parabolic equations. The Runge–Kutta Chebyshev scheme is used for the time integration. The implementation of the proposed method differs from its Eulerian counterpart in the fact that it is applied during each time step, along the characteristic curves rather than in the time direction. The focus is on constructing explicit schemes with a large stability region to solve coupled radiation hydrodynamics models. Numerical results are presented for two test examples in coupled convection-radiation and conduction–radiation problems.

A stabilized explicit Lagrange multiplier based domain decomposition method for parabolic problems

A fully explicit, stabilized domain decomposition method for solving moderately stiff parabolic partial differential equations (PDEs) is presented. Writing the semi-discretized equations as a differential-algebraic equation (DAE) system where the interface continuity constraints between subdomains are enforced by Lagrange multipliers, the method uses the Runge–Kutta–Chebyshev projection scheme to integrate the DAE explicitly and to enforce the constraints by a projection. With mass lumping techniques and node-to-node matching grids, the method is fully explicit without solving any linear system. A stability analysis is presented to show the extended stability property of the method. The method is straightforward to implement and to parallelize. Numerical results demonstrate that it has excellent performance.

S-ROCK: Chebyshev Methods for Stiff Stochastic Differential Equations

We present and analyze a new class of numerical methods for the solution of stiff stochastic differential equations (SDEs). These methods, called S-ROCK (for stochastic orthogonal Runge-Kutta Chebyshev), are explicit and of strong order 1 and possess large stability domains in the mean-square sense. For mean-square stable stiff SDEs, they are much more efficient than the standard explicit methods proposed so far for stochastic problems and give significant speed improvement. The explicitness of the S-ROCK methods allows one to handle large systems without linear algebra problems usually encountered with implicit methods. Numerical results and comparisons with existing methods are reported.

On stabilized integration for time-dependent PDEs

An integration method is discussed which has been designed to treat parabolic and hyperbolic terms explicitly and stiff reaction terms implicitly. The method is a special two-step form of the one-step IMEX (implicit–explicit) RKC (Runge–Kutta–Chebyshev) method. The special two-step form is introduced with the aim of getting a non-zero imaginary stability boundary which is zero for the one-step method. Having a non-zero imaginary stability boundary allows, for example, the integration of pure advection equations space-discretized with centered schemes, the integration of damped or viscous wave equations, the integration of coupled sound and heat flow equations, etc. For our class of methods it also simplifies the choice of temporal step sizes satisfying the von Neumann stability criterion, by embedding a thin long rectangle inside the stability region. Embedding rectangles or other tractable domains with this purpose is an idea of Wesseling.

An adaptive finite element semi-Lagrangian implicit–explicit Runge–Kutta–Chebyshev method for convection dominated reaction–diffusion problems

We introduce in this paper an adaptive method that combines a semi-Lagrangian scheme with a second order implicit–explicit Runge–Kutta–Chebyshev (IMEX RKC) method to calculate the numerical solution of convection dominated reaction–diffusion problems in which the reaction terms are highly stiff. The convection terms are integrated via the semi-Lagrangian scheme, whereas the IMEX RKC treats the diffusion terms explicitly and the highly stiff reaction terms implicitly. The space adaptation is done in the framework of finite elements and the criterion for adaptation is derived from the information supplied by the semi-Lagrangian step; so that, this can be considered a heuristic approach to adaptivity that is somewhat similar to the so-called r-adaptivity strategy.

Runge–Kutta–Chebyshev projection method

In this paper a fully explicit, stabilized projection method called the Runge–Kutta–Chebyshev (RKC) projection method is presented for the solution of incompressible Navier–Stokes systems. This method preserves the extended stability property of the RKC method for solving ODEs, and it requires only one projection per step. An additional projection on the time derivative of the velocity is performed whenever a second-order approximation for the pressure is desired. We demonstrate both by numerical experiments and by order analysis that the method is second order accurate in time for both the velocity and the pressure. Being explicit, the RKC projection method is easy to implement and to parallelize. Hence it is an attractive candidate for the solution of large-scale, moderately stiff, diffusion-like problems.

IRKC: An IMEX solver for stiff diffusion–reaction PDEs

The Fortran 90 code IRKC is intended for the time integration of systems of partial differential equations (PDEs) of diffusion–reaction type for which the reaction Jacobian has real (negative) eigenvalues. It is based on a family of implicit–explicit Runge–Kutta–Chebyshev methods which are unconditionally stable for reaction terms and which impose a stability constraint associated with the diffusion terms that is quadratic in the number of stages. Special properties of the family make it possible for the code to select at each step the most efficient stable method as well as the most efficient step size. Moreover, they make it possible to apply the methods using just a few vectors of storage. A further step towards minimal storage requirements and optimal efficiency is achieved by exploiting the fact that the implicit terms, originating from the stiff reactions, are not coupled over the spatial grid points. Hence, the systems to be solved have a small dimension (viz., equal to the number of PDEs). These characteristics of the code make it especially attractive for problems in several spatial variables. IRKC is a successor to the RKC code [B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: an explicit solver for parabolic PDEs, J. Comput. Appl. Math. 88 (1997) 315–326] that solves similar problems without stiff reaction terms.

Modeling Low Mach Number Reacting Flow with Detailed Chemistry and Transport

An efficient projection scheme is developed for the simulation of reacting flow with detailed kinetics and transport. The scheme is based on a zero-Mach-number formulation of the compressible conservation equations for an ideal gas mixture. It relies on Strang splitting of the discrete evolution equations, where diffusion is integrated in two half steps that are symmetrically distributed around a single stiff step for the reaction source terms. The diffusive half-step is integrated using an explicit single-step, multistage, Runge---Kutta---Chebyshev (RKC) method. The resulting construction is second-order convergent, and has superior efficiency due to the extended real-stability region of the RKC scheme. Two additional efficiency-enhancements are also explored, based on an extrapolation procedure for the transport coefficients and on the use of approximate Jacobian data evaluated on a coarse mesh. We demonstrate the construction in 1D and 2D flames, and examine consequences of splitting errors. By including the above enhancements, performance tests using 2D computations with a detailed C1C2 methane-air mechanism and a mixture-averaged transport model indicate that speedup factors of about 15 are achieved over the starting split-stiff scheme

RKC time-stepping for advection–diffusion–reaction problems

The original explicit Runge–Kutta–Chebyshev (RKC) method is a stabilized second-order integration method for pure diffusion problems. Recently, it has been extended in an implicit–explicit manner to also incorporate highly stiff reaction terms. This implicit–explicit RKC method thus treats diffusion terms explicitly and the highly stiff reaction terms implicitly. The current paper deals with the incorporation of advection terms for the explicit method, thus aiming at the implicit–explicit RKC integration of advection–diffusion–reaction equations in a manner that advection and diffusion terms are treated simultaneously and explicitly and the highly stiff reaction terms implicitly.

An Implicit-Explicit Runge--Kutta--Chebyshev Scheme for Diffusion-Reaction Equations

An implicit-explicit (IMEX) extension of the explicit Runge--Kutta--Chebyshev (RKC) scheme designed for parabolic PDEs is proposed for diffusion-reaction problems with severely stiff reaction terms. The IMEX scheme treats these reaction terms implicitly and diffusion terms explicitly. Within the setting of linear stability theory, the new IMEX scheme is unconditionally stable for reaction terms having a Jacobian matrix with a real spectrum. For diffusion terms the stability characteristics remain unchanged. A numerical comparison for a stiff, nonlinear radiation-diffusion problem between an RKC solver, an IMEX-RKC solver, and the popular implicit BDF solver VODPK using the Krylov solver GMRES illustrates the excellent performance of the new scheme.

A finite element semi-Lagrangian explicit Runge–Kutta–Chebyshev method for convection dominated reaction–diffusion problems

Explicit Runge–Kutta–Chebyshev methods have proved to be efficient for reaction–diffusion problems of moderate stiffness. In this paper, we extend such an efficiency to convection-dominated-reaction–diffusion problems by giving a formulation of these methods in a semi-Lagrangian framework, using C 0-finite elements of degree m⩾2 as the space discretization method. We also study the convergence in the L 2-norm of the methods proposed in this paper.

A numerical study of mixed parabolic–gradient systems

This paper is concerned with the numerical solution of parabolic equations coupled with gradient equations. The gradient equations are ordinary differential equations whose solutions define positions of particles in the spatial domain of the parabolic equations. The vector field of the gradient equations is determined by gradients of solutions to the parabolic equations. Such mixed parabolic–gradient systems are for example, used in neurobiological studies of the formation of axonal connections in the nervous system. We discuss a numerical approach for solving parabolic–gradient systems on a grid. The basic ingredients are the fourth-order spatial finite differencing for the parabolic equations, piecewise cubic Hermite interpolation for approximating the gradient equations, and explicit time-stepping by means of a Runge–Kutta–Chebyshev method.