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.     

Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.     

Unstructured Grid-Based Discontinuous Galerkin Method for Broadband Electromagnetic Simulations

A parallel, unstructured, high-order discontinuous Galerkin method is developed for the time-dependent Maxwell's equations, using simple monomial polynomials for spatial discretization and a fourth-order Runge–Kutta scheme for time marching. Scattering results for a number of validation cases are computed employing polynomials of up to third order. Accurate solutions are obtained on coarse meshes and grid convergence is achieved, demonstrating the capabilities of the scheme for time-domain electromagnetic wave scattering simulations.     

Adaptive Iteration to Steady State of Flow Problems

Runge–Kutta time integration is used to reach the steady state solution of discretized partial differential equations. Continuous and discrete parameters in the method are adapted to the particular problem by minimizing the residual in each step, if this is possible, or the work to reach convergence. Algorithms for parameter optimization are devised and analyzed. Solutions of the linearized Euler equations and the nonlinear Euler and Navier–Stokes equations for compressible flow illustrate the methods.     

Classification of stochastic Runge–Kutta methods for the weak approximation of stochastic differential equations

In the present paper, a class of stochastic Runge–Kutta methods containing the second order stochastic Runge–Kutta scheme due to E. Platen for the weak approximation of Itô stochastic differential equation systems with a multi-dimensional Wiener process is considered. Order 1 and order 2 conditions for the coefficients of explicit stochastic Runge–Kutta methods are solved and the solution space of the possible coefficients is analyzed. A full classification of the coefficients for such stochastic Runge–Kutta schemes of order 1 and two with minimal stage numbers is calculated. Further, within the considered class of stochastic Runge–Kutta schemes coefficients for optimal schemes in the sense that additionally some higher order conditions are fulfilled are presented.     

On the Linear Stability of the Fifth-Order WENO Discretization

We study the linear stability of the fifth-order Weighted Essentially Non-Oscillatory spatial discretization (WENO5) combined with explicit time stepping applied to the one-dimensional advection equation. We show that it is not necessary for the stability domain of the time integrator to include a part of the imaginary axis. In particular, we show that the combination of WENO5 with either the forward Euler method or a two-stage, second-order Runge–Kutta method is linearly stable provided very small time step-sizes are taken. We also consider fifth-order multistep time discretizations whose stability domains do not include the imaginary axis. These are found to be linearly stable with moderate time steps when combined with WENO5. In particular, the fifth-order extrapolated BDF scheme gave superior results in practice to high-order Runge–Kutta methods whose stability domain includes the imaginary axis. Numerical tests are presented which confirm the analysis.     

A Chebyshev pseudospectral multidomain method for the soliton solution of coupled nonlinear Schrödinger equations

The collision of solitary waves is an important problem in both physics and applied mathematics. In this paper, we study the solution of coupled nonlinear Schrödinger equations based on pseudospectral collocation method with domain decomposition algorithm for approximating the spatial variable. The problem is converted to a system of nonlinear ordinary differential equations which will be integrated in time by explicit Runge–Kutta method of order four. The multidomain scheme has much better stability properties than the single domain. Thus this permits using much larger step size for the time integration which fulfills stability restrictions. The proposed scheme reduces the effects of round-of-error for the Chebyshev collocation and also uses less memory without sacrificing the accuracy. The numerical experiments are presented which show the multidomain pseudospectral method has excellent long-time numerical behavior and preserves energy conservation property.     

Numerical simulation of stochastic replicator models in catalyzed RNA-like polymers

A stochastic model for replicators in catalyzed RNA-like polymers is presented and numerically solved. The model consists of a system of reaction–diffusion equations describing the evolution of a population formed by RNA-like molecules with catalytic capabilities in a prebiotic process. The diffusion effects and the catalytic reactions are deterministic. A stochastic excitation with additive noise is introduced as a force term. To numerically solve the governing equations we apply the stochastic method of lines. A finite-difference reaction–diffusion system is constructed by discretizing the space and the associated stochastic differential system is numerically solved using a class of stochastic Runge–Kutta methods. Numerical experiments are carried out on a prototype of four catalyzed selfreplicator species along with an activated and an inactivated residues. Results are given in two space dimensions.     

High-order Compact Schemes for Nonlinear Dispersive Waves

High-order compact finite difference schemes coupled with high-order low-pass filter and the classical fourth-order Runge–Kutta scheme are applied to simulate nonlinear dispersive wave propagation problems described the Korteweg-de Vries (KdV)-like equations, which involve a third derivative term. Several examples such as KdV equation, and KdV-Burgers equation are presented and the solutions obtained are compared with some other numerical methods. Computational results demonstrate that high-order compact schemes work very well for problems involving a third derivative term.     

Toward Non Commutative Numerical Analysis: High Order Integration in Time

We review some methods for high precision time integration: it is not easy to ensure stability, precision and numerical efficiency at the same time. Operator splitting—when it works—can be a good way to satisfy all these constraints; in some cases, the order of the splitting schemes can be enhanced by extrapolation; nevertheless, the applicability of splitting is limited due to non commutativity. As an alternative to splitting, we introduce preconditioned Runge–Kutta (PRK) schemes: the preconditioning is included in the scheme, instead of being put aside for implementation. Examples of PRK schemes are given including the extrapolation of the residual smoothing scheme, and sufficient conditions for stability are described.     

Correction of Numerical Integration as an Optimal Control Problem

The drift of numerical solution of a dynamical system off the integral and constraint surface is eliminated with the help of a gradient supplement to the equations of motion. This supplement makes the image point in the phase space to move along the normal to a given surface. The solution is compared with those obtained by integrating the generalized Hamilton–Dirac dynamics equations. The calculations were performed in Maple V R5 with the use of the standard subroutine dsolve/numeric for solving ordinary differential equations by the fourth-order Runge–Kutta method.     

Exponential Runge–Kutta methods for delay differential equations

This paper deals with convergence and stability of exponential Runge–Kutta methods of collocation type for delay differential equations. It is proved that these kinds of numerical methods converge at least with their stage order. Moreover, a sufficient condition of the numerical stability is provided. Finally, some numerical examples are presented to illustrate the main conclusions.     

Construction and Implementation of General Linear Methods for Ordinary Differential Equations: A Review

It it the purpose of this paper to review the results on the construction and implementation of diagonally implicit multistage integration methods for ordinary differential equations. The systematic approach to the construction of these methods with Runge–Kutta stability is described. The estimation of local discretization error for both explicit and implicit methods is discussed. The other implementations issues such as the construction of continuous extensions, stepsize and order changing strategy, and solving the systems of nonlinear equations which arise in implicit schemes are also addressed. The performance of experimental codes based on these methods is briefly discussed and compared with codes from Matlab ordinary differential equation (ODE) suite. The recent work on general linear methods with inherent Runge–Kutta stability is also briefly discussed     

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     

Study of a third grade non-Newtonian fluid flow between two parallel plates using the multi-step differential transform method

In this paper, the multi-step differential transform method (MDTM), one of the most effective method, is implemented to compute an approximate solution of the system of nonlinear differential equations governing the problem. It has been attempted to show the reliability and performance of the MDTM in comparison with the numerical method (fourth-order Runge–Kutta) and other analytical methods such as HPM, HAM and DTM in solving this problem. The first differential equation is the plane Couette flow equation which serves as a useful model for many interesting problems in engineering. The second one is the Fully-developed plane Poiseuille flow equation and finally the third one is the plane Couette–Poiseuille flow.     

High Order Numerical Discretization for Hamilton–Jacobi Equations on Triangular Meshes

In this paper we construct several numerical approximations for first order Hamilton–Jacobi equations on triangular meshes. We show that, thanks to a filtering procedure, the high order versions are non-oscillatory in the sense of satisfying the maximum principle. The methods are based on the first order Lax–Friedrichs scheme [2] which is improved here adjusting the dissipation term. The resulting first order scheme is -monotonic (we explain the expression in the paper) and converges to the viscosity solution as for the L -norm. The first high order method is directly inspired by the ENO philosophy in the sense where we use the monotonic Lax–Friedrichs Hamiltonian to reconstruct our numerical solutions. The second high order method combines a spatial high order discretization with the classical high order Runge–Kutta algorithm for the time discretization. Numerical experiments are performed for general Hamiltonians and L ¹, L ² and L -errors with convergence rates calculated in one and two space dimensions show the k-th order rate when piecewise polynomial of degree k functions are used, measured in L ¹-norm.     

A Comparative Study of Time Advancement Methods for Solving Navier–Stokes Equations

A qualitative and quantitative study is made for choosing time advancement strategies for solving time dependent equations accurately. A single step, low order Euler time integration method is compared with Adams–Bashforth, a second order accurate time integration strategy for the solution of one dimensional wave equation. With the help of the exact solution, it is shown that the presence of the computational mode in Adams–Bashforth scheme leads to erroneous results, if the solution contains high frequency components. This is tested for the solution of incompressible Navier–Stokes equation for uniform flow past a rapidly rotating circular cylinder. This flow suffers intermittent temporal instabilities implying presence of high frequencies. Such instabilities have been noted earlier in experiments and high accuracy computations for similar flow parameters. This test problem shows that second order Adams– Bashforth time integration is not suitable for DNS.     

A study of suspended roofs for near-source seismic motions

The non-linear behavior of multi-suspended roof systems for seismic loads is studied. The study is based on a formulation that can be easily employed for a preliminary design of multi-suspended roofs subjected to seismic loads. Specifically, applying Lagrange’s equations, the corresponding set of equations of motion for discrete models of multiple suspension roofs is obtained and numerical integration of the equations of motion is performed via the Runge–Kutta scheme. For representative realistic combinations of geometric, stiffness and damping parameters, a non-linear analysis is employed to study the behavior of suspended roofs for near-source and far-field seismic motions. The analysis demonstrates that: (i) code-specified design loads could dramatically underestimate the response of suspended roofs subjected to near-source ground motions and (ii) flexible roofing systems are greatly affected by near-source ground motions, a behavior that is not observed for stiff systems.     

Level Set Based Simulations of Two-Phase Oil–Water Flows in Pipes

We simulate the axisymmetric pipeline transportation of oil and water numerically under the assumption that the densities of the two fluids are different and that the viscosity of the oil core is very large. We develop the appropriate equations for core-annular flows using the level set methodology. Our method consists of a finite difference scheme for solving the model equations, and a level set approach for capturing the interface between two liquids (oil and water). A variable density projection method combined with a TVD Runge–Kutta scheme is used to advance the computed solution in time. The simulations succeed in predicting the spatially periodic waves called bamboo waves, which have been observed in the experiments of [Bai et al. (1992) J. Fluid Mech. 240, 97–142.] on up-flow in vertical core flow. In contrast to the stable case, our simulations succeed in cases where the oil breaks up in the water, and then merging occurs. Comparisons are made with other numerical methods and with both theoretical and experimental results.