A Stability Criterion for Semi-Discrete Difference Schemes of Hyperbolic Conservation Laws on Uniform Grids

In the present study, the stability condition for semi-discrete difference schemes of hyperbolic conservation laws obtained from Fourier analysis is simplified. This stability condition can be applied only to linear difference schemes with constant coefficients implemented with periodic boundary treatment. It could often give useful results for other cases, such as schemes with variable coefficients, schemes for nonperiodic problem and nonlinear problem. However, this condition usually leads to a trigonometric inequality, which makes it not convenient to use. For explicit difference schemes on uniform grids, this trigonometric inequality can be converted to polynomial form. Furthermore, if the scheme is a high-order one, the polynomial can be factorized into a simple form. Thus, it is much easier to solve than the inequality obtained directly from Fourier analysis. For compact difference schemes and conservative schemes, similar results are obtained. Some applications of this new stability criterion are shown, including judging the stability of two schemes, proving the upstream central schemes to be stable, constructing a stable upwind dissipation relation preserving (DRP) scheme and constructing an optimized weighted essentially non-oscillatory (WENO) scheme. Since WENO schemes are nonlinear schemes, the stability analysis in the present study is performed on their underlying linear schemes. According to the numerical tests, the underlying linear scheme should be stable, otherwise the corresponding WENO scheme may display instability. These applications demonstrate that this criterion is convenient and efficient for judging the linear stability of semi-discrete difference schemes and constructing stable upwind difference schemes.     

Numerical Anisotropy Study of a Class of Compact Schemes

We study the numerical anisotropy existent in compact difference schemes as applied to hyperbolic partial differential equations, and propose an approach to reduce this error and to improve the stability restrictions based on a previous analysis applied to explicit schemes. A prefactorization of compact schemes is applied to avoid the inversion of a large matrix when calculating the derivatives at the next time level, and a predictor–corrector time marching scheme is used to update the solution in time. A reduction of the isotropy error is attained for large wave numbers and, most notably, the stability restrictions associated with MacCormack time marching schemes are considerably improved. Compared to conventional compact schemes of similar order of accuracy, the multidimensional schemes employ larger stencils which would presumably demand more processing time, but we show that the new stability restrictions render the multidimensional schemes to be in fact more efficient, while maintaining the same dispersion and dissipation characteristics of the one dimensional schemes     

Stability analysis of six-point finite difference schemes for the constant coefficient convective-diffusion equation

A comprehensive and systematic study is presented to derive stability properties of various two-level, six-point finite difference schemes (in particular, difference schemes of Padé type) for the approximation to the constant coefficient convective-diffusion equation. First, the modified equivalent partial differential equation (MEPDE) for a general six-point difference scheme is derive. The MEPDE provides direct information on the order of accuracy of a difference scheme. The von Neumann and matrix methods are then employed to deduce the necessary and sufficient conditions for the numerical stability for the six-point difference schemes. An unified technique is developed to find the stability regions for the difference schemes. Some new second and third order six-point difference schemes for the approximation of the constant coefficient convective-diffusion equation are presented.     

On the Non-linear Stability of Flux Reconstruction Schemes

The flux reconstruction (FR) approach unifies various high-order schemes, including collocation based nodal discontinuous Galerkin (DG) methods, and all spectral difference methods (at least for a linear flux function), within a single framework. Recently a new range of linearly stable FR schemes have been identified, henceforth referred to as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes. In this short note non-linear stability properties of FR schemes are elucidated via analysis of linearly stable VCJH schemes (so as to focus attention solely on issues of non-linear stability). It is shown that linearly stable VCJH schemes (at least in their standard form) may be unstable if the flux function is non-linear. This instability is due to aliasing errors, which manifest since FR schemes (in their standard form) utilize a collocation projection at the solution points to construct a polynomial approximation of the flux. Strategies for minimizing such aliasing driven instabilities are discussed within the context of the FR approach. In particular, it is shown that the location of the solution points will have a significant effect on non-linear stability. This result is important, since linear analysis of FR schemes implies stability is independent of solution point location. Finally, it is shown that if an exact L2 projection is employed to construct an approximation of the flux (as opposed to a collocation projection), then aliasing errors and hence aliasing driven instabilities will be eliminated. However, performing such a projection exactly, or at least very accurately, would be more costly than performing a collocation projection, and would certainly impact the inherent efficiency and simplicity of the FR approach. It can be noted that in all above regards, non-linear stability properties of FR schemes are similar to those of nodal DG schemes. The findings should motivate further research into the non-linear performance of FR schemes, which have hitherto been developed and analyzed solely in the context of a linear flux function.     

Finite difference methods for viscous incompressible global stability analysis

In the present article some high-order finite-difference schemes and in particularly dispersion-relation-preserving (DRP) family schemes, initially developed by Tam and Webb [Dispersion-relation-preserving finite difference schemes for computational acoustics, J. Comput. Phys. 107 (1993) 262-281.] for computational aeroacoustic problems, are used for global stability issue. (The term global is not used in weakly-non-parallel framework but rather for fully non-parallel flows. Some authors like Theofilis [Advances in global linear instability analysis of non-parallel and three-dimensional flows, Progress in Aerospace Sciences 39 (2003) 249-315] refer to this approach as “BiGlobal”.) These DRP schemes are compared with different classical schemes as second and fourth-order finite-difference schemes, seven-order compact schemes and spectral collocation scheme which is usually employed in such stability problems. A detailed comparative study of these schemes for incompressible flows over two academic configurations (square lid-driven cavity and separated boundary layer at different Reynolds numbers) is presented, and we intend to show that these schemes are sufficiently accurate to perform global stability analyses.     

Stability analysis and robust control for fixed-scale teleoperation

In this article we present a new theorem that reduces the stability problem of teleoperators from a structured singular value problem to a maximum singular value problem. A control scheme realizing an ideal response for fixed-scale teleoperation is proposed and its stability is proven analytically. Next, the ideal control scheme is extended to control schemes with a higher stability robustness. Also, for these more complicated schemes, stability analyses are performed making use of the new theorem and stability conditions are derived. Experiments on a 1-d.o.f. setup are conducted to confirm the validity of the proposed control schemes.     

The stability of explicit difference schemes for solving the problem of interaction between a compressible fluid and an elastic shell

A mathematical model for describing the interaction between a compressible fluid and an elastic shell is formulated as an initial boundary value problem. The partial differential equations of the model are discretized both in time and space by a finite-difference method. The stability of the resulting explicit difference schemes is analyzed by Kreiss' theory for the stability analysis of difference schemes in initial boundary value problems. It is shown that the stability properties of the schemes for the interaction problem may be influenced by the type of discretization in space used for the contact condition on the interface between the fluid and the shell and also by the approximation of the hydrodynamic pressure on the surface of the shell. A simple sufficient condition is found that will ensure the best possible stability properties of the schemes. Several of these, which are of practical interest, are analyzed.     

Stability of weak numerical schemes for stochastic differential equations

This paper considers numerical stability and convergence of weak schemes solving stochastic differential equations. A relatively strong notion of stability for a special type of test equations is proposed. These are stochastic differential equations with multiplicative noise. For different explicit and implicit schemes, the regions of stability are also examined.     

Euler-like discrete models of the logistic differential equation

To understand the behavior of difference schemes on nonlinear differential equations, it seems desirable to extend the standard linear stability theory into a nonlinear theory. As a step in that direction, we investigate the stability properties of Euler-related integration algorithms by checking how they preserve and violate the dynamical structure of the logistic differential equation.Among the schemes considered are two linearly implicit nonstandard schemes which are adjoint to each other. We find that these schemes are superior to explicit schemes when they are stable and the blow-up time has not passed: for these λh-values they are dynamically faithful. When these schemes ‘turn unstable’, however, they have much less desirable properties than explicit or fully implicit schemes: they become simultaneously superstable and unstable. This is explained by the fact that these schemes are not self-adjoint: the linearly implicit self-adjoint scheme is dynamically faithful in an Euler-typical range of step sizes and gives correct stability for all step sizes.     

Practical aspects of numerical time integration

This paper examines some practical aspects in the numerical time integration of structural equations of motion. The concept of stability region in the complex frequency/time-step (ω · Δt) plane is proposed as the basis of stability evaluation, as opposed to the concept of conditional (or unconditional) stability. Stabilization of numerical computations via the introduction of artificial damping and the composition of existing schemes is discussed. It is shown that stabilization by artificial damping fails for any explicit scheme contrary to the notion that a frequency-proportional damping would suppress the high-frequency components. It is also shown that among explicit schemes examined in this paper, the central difference scheme is preferred from the stability considerations. For nonlinear problems, the linearly extrapolated pseudo-force solution procedure is adopted to assess how nonlinearities affect the stability of implicit schemes. Finally, the impact of stability and accuracy characteristics upon numerical computation is discussed.     

A class of linearized energy-conserved finite difference schemes for nonlinear space-fractional Schrödinger equations

In this work, we propose a class of linearized energy-conserved finite difference schemes for nonlinear space-fractional Schrödinger equations. We prove the energy conservation, stability, and convergence of our schemes. In the proposed schemes, we only need to solve linear algebraic systems to obtain the numerical solutions. Numerical examples are presented to verify the accuracy, energy conservation, and stability of these schemes.     

A family of stable numerical solvers for the shallow water equations with source terms

In this work we introduce a multiparametric family of stable and accurate numerical schemes for 1D shallow water equations. These schemes are based upon the splitting of the discretization of the source term into centered and decentered parts. These schemes are specifically designed to fulfill the enhanced consistency condition of Bermúdez and Vázquez, necessary to obtain accurate solutions when source terms arise. Our general family of schemes contains as particular cases the extensions already known of Roe and Van Leer schemes, and as new contributions, extensions of Steger–Warming, Vijayasundaram, Lax–Friedrichs and Lax–Wendroff schemes with and without flux-limiters. We include some meaningful numerical tests, which show the good stability and consistency properties of several of the new methods proposed. We also include a linear stability analysis that sets natural sufficient conditions of stability for our general methods.     

On the computation of two-dimensional compressible flow fields by the modified local stability scheme

The modified local stability scheme is applied to several two-dimensional problems—blunt body flow, regular reflection of a shock and lambda shock. The resolution of the flow features obtained by the modified local stability scheme is found to be better than that achieved by the other first order schemes and almost identical to that achieved by the second order schemes incorporating artificial viscosity. The scheme is easy for coding, consumes moderate amount of computer storage and time. The scheme can be advantageously used in place of second order schemes.     

Unconditionally positivity and boundedness preserving schemes for a FitzHugh–Nagumo equation

In this paper, a semi-explicit scheme is constructed for the space-independent FitzHugh–Nagumo equation. Qualitative stability analysis shows that the semi-explicit scheme is dynamically consistent with the space independent equation. Then, the semi-explicit scheme is extended to construct a new finite difference scheme for the full FitzHugh–Nagumo equation in one- and two-space dimensions, respectively. According to the theory of M-matrices, it is proved that these new schemes are able to preserve the positivity and boundedness of solutions of the corresponding equations for arbitrary step sizes. The consistency and numerical stability of these schemes is also analysed. Combined with the property of the strictly diagonally dominant matrix, the convergence of these schemes is established. Numerical experiments illustrate our results and display the advantages of our schemes in comparison to some other schemes.     

Unconditionally stable high-order time integration for moving mesh finite difference solution of linear convection–diffusion equations

This paper is concerned with moving mesh finite difference solution of partial differential equations. It is known that mesh movement introduces an extra convection term and its numerical treatment has a significant impact on the stability of numerical schemes. Moreover, many implicit second and higher order schemes, such as the Crank–Nicolson scheme, will lose their unconditional stability. A strategy is presented for developing temporally high-order, unconditionally stable finite difference schemes for solving linear convection–diffusion equations using moving meshes. Numerical results are given to demonstrate the theoretical findings.     

A new way for constructing high accuracy shock-capturing generalized compact difference schemes

The generalized compact (GC) schemes and some of their important properties are presented. And a new way for constructing high order accuracy and high-resolution GC schemes is presented. The schemes constructed by using this way could satisfy some principles and demands prescribed in advance to ensure some desired properties to the schemes, such as the principle about suppression of the oscillations, the principle of stability, the order of accuracy and number of scheme points, etc. As two examples, a three-point third-order compact scheme and a three-point fifth-order GC scheme satisfying the principle about suppression of the oscillations and the principle of stability are described in this paper. Numerical results show that these schemes are shock-capturing. The time-dependent boundary conditions proposed by Thompson are well employed when the algorithm is applied to the Euler equations of gas dynamics. Fourier analysis shows that the resolution characteristics are spectral-like.     

Time Explicit Schemes and Spatial Finite Differences Splittings

In this article, we conjugate time marching schemes with Finite Differences splittings into low and high modes in order to build fully explicit methods with enhanced temporal stability for the numerical solutions of PDEs. The main idea is to apply explicit schemes with less restrictive stability conditions to the linear term of the high modes equation, in order that the allowed time step for the temporal integration is only determined by the low modes. These conjugated schemes were developed in [10] for the spectral case and here we adapt them to the Finite Differences splittings provided by Incremental Unknowns, which steems from the Inertial Manifolds theory. We illustrate their improved capabilities with numerical solutions of Burgers equations, with uniform and nonuniform meshes, in dimensions one and two, when using modified Forward–Euler and Adams–Bashforth schemes. The resulting schemes use time steps of the same order of those used by semi-implicit schemes with comparable accuracy and reduced computational costs.     

Stability of Some Generalized Godunov Schemes With Linear High-Order Reconstructions

Classical Semi-Lagrangian schemes have the advantage of allowing large time steps, but fail in general to be conservative. Trying to keep the advantages of both large time steps and conservative structure, Flux-Form Semi-Lagrangian schemes have been proposed for various problems, in a form which represent (at least in a single space dimension) a high-order, large time-step generalization of the Godunov scheme. At the theoretical level, a recent result has shown the equivalence of the two versions of Semi-Lagrangian schemes for constant coefficient advection equations, while, on the other hand, classical Semi-Lagrangian schemes have been proved to be strictly related to area-weighted Lagrange–Galerkin schemes for both constant and variable coefficient equations. We address in this paper a further issue in this theoretical framework, i.e., the relationship between stability of classical and of Flux-Form Semi-Lagrangian schemes. We show that the stability of the former class implies the stability of the latter, at least in the case of the one-dimensional linear continuity equation.     

Computations with inverse Runge-Kutta schemes

A new subclass of schemes is considered formally reduced to the class of fully implicit Runge-Kutta schemes possessing outstanding accuracy and stability characteristics. The implementation details of the iterative algorithm for solving stiff systems of ODE and differential-algebraic systems of index 1 by means of the proposed schemes are given.     

On two upwind finite-difference schemes for hyperbolic equations in non-conservative form

Several finite difference-schemes for approximating solutions of initial value problems associated with systems of linear hyperbolic differential equations are considered. Common features of the schemes is the approximation of the space-like derivatives according to the behavior of the characteristics (upwind schemes for hyperbolic equations). The analysis of standard properties (consistency, stability, convergence, dissipativity, phase error) of finite difference schemes is performed. In addition, extensions of certain upwind schemes to nonlinear equations, extensions to several space-like dimensions by splitting methods and two implicit finite difference schemes are considered.