Monotone bicompact schemes for a linear advection equation

Doklady Mathematics -     

Comparison of numerical schemes for solving the advection equation

We report on the dispersion and dissipation properties of numerical schemes aimed at solving the one-dimensional advection equation. The study is based on the consistency error, which is explicitly calculated for various standard finite-difference schemes. The oscillation and damping features of the numerical solutions are shown to be explained via a generalized Airy-like function. In the specific case of the advection of a step function, the solutions of the equivalent equations are systematically calculated and shown to recover the numerical solutions. A particular emphasis is put on one third-order accurate scheme, which involves a weak smearing of the step.     

On bicompact grid-characteristic schemes for the linear advection equation

A set of grid-characteristic schemes for the linear advection equation is considered. Depending on the behavior of the solution, hybrid compact difference schemes of second–third order accuracy are proposed as based on interpolation polynomials. The schemes produce monotone solutions and only slightly smear discontinuities.     

Fourth-order diagonally implicit schemes for evolution equations

New fourth-order methods are proposed for solving both ordinary and partial differential equations. The derivation of the methods is based on the form of diagonally implicit schemes applied to stiff ordinary differential equations. The methods are absolutely and unconditionally stable. Test computations are presented.     

Parallel iteration schemes for implicit ODEIVP methods

In this contribution to the Proceedings of the “International Symposium on New Aspects of Numerical Analysis in the Light of Recent Technology” held at Stresa on 13 until 18 September 1993, we give a survey of recent research at CWI for solving the implicit relations arising in ODEIVP methods on parallel computers. Starting with a General Linear method as introduced by Butcher, three forms of parallelism for solving the associated implicit equations are discussed, viz. (i) parallelism across the stages within a single step (stage parallelism), (ii) parallel preconditioners, and (iii) parallelism across the steps (step parallelism). The structure of these type of parallel methods will be described.     

Explicit finite difference schemes with extended stability for advection equations

In this study an explicit central difference approximation of the generalized leap-frog type is applied to the one- and two-dimensional advection equations. The stability of the considered numerical schemes is investigated and the scheme with the largest stable time step is found. For the linear and nonlinear advection equations numerical experiments with different schemes from the considered class are performed in order to evaluate the practical stability of the designed schemes.     

Finding nonoscillatory solutions to difference schemes for the advection equation

The advection equation is solved using a weighted adaptive scheme that combines a monotone scheme with the central-difference approximation of the first spatial derivative. The determination of antidiffusion fluxes is treated as an optimization problem. The solvability of the optimization problem is analyzed, and the differential properties of the cost functional are examined. It is shown that the determination of antidiffusion fluxes is reduced to a linear programming problem in the case of an explicit scheme and to a nonlinear programming problem or a sequence of linear programming problems in the case of an implicit scheme. A simplified monotonization algorithm is proposed. Numerical results are presented.     

High-order accurate implicit running schemes

An approach to the construction of high-order accurate implicit predictor-corrector schemes is proposed. The accuracy is improved by choosing a special time integration step for computing numerical fluxes through cell interfaces by using an unconditionally stable implicit scheme. For smooth solutions of advection equations with constant coefficients, the scheme is second-order accurate. Implicit difference schemes for multidimensional advection equations are constructed on the basis of Godunov’s method with splitting over spatial variables as applied to the computation of “large” values at an intermediate layer. The numerical solutions obtained for advection equations and the radiative transfer equations in a vacuum are compared with their exact solutions. The comparison results confirm that the approach is efficient and that the accuracy of the implicit predictor-corrector schemes is improved.     

Rational approximations by implicit Runge-Kutta schemes

Ehle [3] has pointed out that then-stage implicit Runge-Kutta (IRK) methods due to Butcher [1] areA-stable in the definition of Dahlquist [2] because they effect the operationR(Ah) whereR(μ) is the diagonal Padé approximation toe ^µ. The purpose of this note is to point out that ifR(μ)=P(μ)/Q(μ) is a rational polynomial whosen poles are distinct and nonzero, and if degreeP(μ)≦degreeQ(μ)=n, then ann-stage IRK method applied toy=A y can be used for the operation $$y^{n + 1} = R(Ah)y^n $$ This will no longer be of order 2n, nor necessarily the same order as the approximation ofR(Ah) toe ^Ah. However, if any particularly useful integration formsR can be found, they can be performed by the IRK method.     

Stability of a family of multi-time-level difference schemes for the advection equation

Summary For the linear advection equation we consider explicit multi-time-level schemes of highest order which are one step in space direction only. If a stencil involvesk time steps we show that it is stable in theL ₂-sense for Courant numbers in the interval (0, 1/k). Since the order is 2k–1 one can use these schemes for high order discretization of the boundary conditions in hyperbolic initial value problems.Part of this work has been performed in the project Mehrschritt-Differenzenschemata of the Schwerpunktprogramm Finite Approximationen in der Strömungsmechanik which has been supported by the DFG     

Stability of implicit schemes on nonuniform grids

Implicit finite-difference approximations of the quasilinear conservation law are considered. Issues of stability and convergence are discussed, and an accuracy bound is obtained.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 157–161.     

Some linear stability results for iterative schemes for implicit Runge-Kutta methods

This article examines stability properties of some linear iterative schemes that have been proposed for the solution of the nonlinear algebraic equations arising in the use of implicit Runge-Kutta methods to solve a differential systemx =f(x). Each iteration step requires the solution of a set of linear equations, with constant matrixI –hJ, whereJ is the Jacobian off evaluated at some fixed point. It is shown that the stability properties of a Runge-Kutta method can be preserved only if is an eigenvalue of the coefficient matrixA. SupposeA has minimal polynomial (x – ) ^m p(x),p() 0. Then stability can be preserved only if the order of the method is at mostm + 2 (at mostm + 1 except for one case).This work was partially supported by a grant from the Science and Engineering Research Council.     

Modification of basis functions in high order discontinuous Galerkin schemes for advection equation

High order discontinuous Galerkin (DG) discretization schemes are considered for an advection boundary-value problem on 2-D unstructured grids with arbitrary geometry of grid cells. A number of test cases are developed to study the sensitivity of a high order DG scheme to local grid distortion. It will be demonstrated how to modify the formulation of a DG discretization for the advection equation. Our approach allows one to maintain the required accuracy on distorted grids while using a fewer number of basis functions for the solution approximation in order to save computational resources.     

Fast implicit schemes for the Fokker–Planck–Landau equation

We propose new implicit schemes to solve the homogeneous and isotropic Fokker–Planck–Landau equation. These schemes have conservation and entropy properties. Moreover, they allow for large time steps (of the order of the physical relaxation time), contrary to usual explicit schemes. We use in particular fast linear Krylov solvers like the GMRES method. These schemes allow an important gain in terms of CPU time, with the same accuracy as explicit schemes. This work is a first step to the development of fast implicit schemes to solve more realistic kinetic models. To cite this article: M. Lemou, L. Mieussens, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Reducing and monitoring round-off error propagation for symplectic implicit Runge-Kutta schemes

We propose an implementation of symplectic implicit Runge-Kutta schemes for highly accurate numerical integration of non-stiff Hamiltonian systems based on fixed point iteration. Provided that the computations are done in a given floating point arithmetic, the precision of the results is limited by round-off error propagation. We claim that our implementation with fixed point iteration is near-optimal with respect to round-off error propagation under the assumption that the function that evaluates the right-hand side of the differential equations is implemented with machine numbers (of the prescribed floating point arithmetic) as input and output. In addition, we present a simple procedure to estimate the round-off error propagation by means of a slightly less precise second numerical integration. Some numerical experiments are reported to illustrate the round-off error propagation properties of the proposed implementation.     

Three-and four-step implicit absolutely stable fourth-order Runge-Kutta schemes

Two types of implicit fourth-order Runge-Kutta schemes are constructed for first-order ordinary differential equations, multidimensional transfer equations, and compressible gas equations. The absolute stability of the schemes is proved by applying the principle of frozen coefficients. Adaptive artificial viscosity ensuring good time convergence and oscillations damping near discontinuities is used in solving gas dynamics equations. The comparative efficiency of the schemes is illustrated by numerical results obtained for compressible gas flows.     

Stability and error analysis on partially implicit schemes

Subdomain techniques have been widely used for solving elliptic and parabolic equations. For parabolic problems, it is possible to combine subdomain techniques with explicit methods to construct efficient algorithms. In addition, this kind of algorithms is naturally suitable for parallel computing. However, the stability of such schemes has been considered as a very difficult issue. In this article, we use an exact error propagation and discrete scheme smoothing approach to give a posteriori stability and error analysis. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Analysis of a class of quasi-monotone and conservative semi-Lagrangian advection schemes

Summary.   We analyze in the norm a class of semi-Lagrangian advective schemes introduced by the author and A. Staniforth in 1992 to improve the solution produced by numerical models for weather prediction and climate studies that use semi-Lagrangian advective schemes. The new quasi-monotone and conservative semi-Lagrangian schemes are stable and converge optimally when the solution is sufficiently smooth. Received May 17, 1999 / Revised version received November 22, 1999 / Published online August 24, 2000     

Row-ordering schemes for sparse givens transformations. II. implicit graph model

A new graph model is presented to study the row annihilation and row ordering problems in the QR decomposition of sparse matrices using Givens rotations. The graph-theoretic results obtained can be used to derive good row orderings for certain column orderings, such as width-1 and width-2 nested dissection orderings. This model is different from the bipartite-graph model introduced in [6]. We refer to the new model as implicit because the rows are not represented explicitly by nodes, in contrast to the bipartite-graph model, where the rows are represented by nodes in a bipartite graph.     

Explicit and implicit projection-difference schemes for the solution of the Navier-Stokes equations

Journal of Mathematical Sciences -