Stability Preserving Research Articles

A New Algorithm for Model Order Reduction of Interval Systems

Mixed method of interval systems is a combination of classical reduction methods and stability preserving methods of interval systems. This paper proposed a new method for model order reduction of systems with uncertain parameters. The bounds on the uncertain parameters are known a priori. Two separate methods are used for finding parameters of the numerator and denominator. The numerator parameters are obtained by either of these methods such as differentiation method, factor division method, cauer second form, moment matching method or Pade approximation method. The denominator is obtained by the differentiation method in all the cases. A numerical example has been discussed to illustrate the procedures. From the above mixed methods, differentiation method and cauer second form as resulted in better approximation when compared with other methods. The errors between the original higher order and reduced order models have also been highlighted to support the effectiveness of the proposed methods.

Strong Stability Preserving Explicit Runge--Kutta Methods of Maximal Effective Order

We apply the concept of effective order to strong stability preserving (SSP) explicit Runge-Kutta methods. Relative to classical Runge-Kutta methods, methods with an effective order of accuracy are designed to satisfy a relaxed set of order conditions, but yield higher order accuracy when composed with special starting and stopping methods. We show that this allows the construction of four-stage SSP methods with effective order four (such methods cannot have classical order four). However, we also prove that effective order five methods - like classical order five methods - require the use of non-positive weights and so cannot be SSP. By numerical optimization, we construct explicit SSP Runge-Kutta methods up to effective order four and establish the optimality of many of them. Numerical experiments demonstrate the validity of these methods in practice.

Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection–diffusion equations on triangular meshes

We propose second order accurate discontinuous Galerkin (DG) schemes which satisfy a strict maximum principle for general nonlinear convection–diffusion equations on unstructured triangular meshes. Motivated by genuinely high order maximum-principle-satisfying DG schemes for hyperbolic conservation laws (Perthame, 1996) and (Zhang, 2010) [14,26], we prove that under suitable time step restriction for forward Euler time stepping, for general nonlinear convection–diffusion equations, the same scaling limiter coupled with second order DG methods preserves the physical bounds indicated by the initial condition while maintaining uniform second order accuracy. Similar to the purely convection cases, the limiters are mass conservative and easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. Following the idea in Zhang (2012) [30], we extend the schemes to two-dimensional convection–diffusion equations on triangular meshes. There are no geometric constraints on the mesh such as angle acuteness. Numerical results including incompressible Navier–Stokes equations are presented to validate and demonstrate the effectiveness of the numerical methods.

Total-variation-diminishing implicit–explicit Runge–Kutta methods for the simulation of double-diffusive convection in astrophysics

We put forward the use of total-variation-diminishing (or more generally, strong stability preserving) implicit–explicit Runge–Kutta methods for the time integration of the equations of motion associated with the semiconvection problem in the simulation of stellar convection. The fully compressible Navier–Stokes equation, augmented by continuity and total energy equations, and an equation of state describing the relation between the thermodynamic quantities, is semi-discretized in space by essentially non-oscillatory schemes and dissipative finite difference methods. It is subsequently integrated in time by Runge–Kutta methods which are constructed such as to preserve the total variation diminishing (or strong stability) property satisfied by the spatial discretization coupled with the forward Euler method. We analyse the stability, accuracy and dissipativity of the time integrators and demonstrate that the most successful methods yield a substantial gain in computational efficiency as compared to classical explicit Runge–Kutta methods.

Implementation of an X-FEM Solver for the Classical Two-Phase Stefan Problem

The classical two-phase Stefan problem in level set formulation is considered. The implementation of a solver on triangular grids is described. Extended finite elements (X-FEM) in space and an implicit Euler method in time are used to approximate the temperature. For the level set equation, a discontinuous Galerkin (DG) and a strong stability preserving (SSP) Runge-Kutta scheme are employed. Polynomial spaces of quadratic order are used. A numerical example with a change of topology is provided, and the order of convergence is studied on the Frank sphere example.

Special boundedness properties in numerical initial value problems

For Runge-Kutta methods, linear multistep methods and other classes of general linear methods much attention has been paid in the literature to important nonlinear stability properties known as total-variation-diminishing (TVD), strong stability preserving (SSP) and monotonicity. Stepsize conditions guaranteeing these properties were studied by Shu and Osher (J. Comput. Phys. 77:439–471, 1988) and in numerous subsequent papers. Unfortunately, for many useful methods it has turned out that these properties do not hold. For this reason attention has been paid in the recent literature to the related and more general properties called total-variation-bounded (TVB) and boundedness. In the present paper we focus on stepsize conditions guaranteeing boundedness properties of a special type. These boundedness properties are optimal, and distinguish themselves also from earlier boundedness results by being relevant to sublinear functionals, discrete maximum principles and preservation of nonnegativity. Moreover, the corresponding stepsize conditions are more easily verified in practical situations than the conditions for general boundedness given thus far in the literature. The theoretical results are illustrated by application to the two-step Adams-Bashforth method and a class of two-stage multistep methods.

On Padé approximations, quadratic stability and discretization of switched linear systems

In this note we consider the stability preserving properties of diagonal Padé approximations to the matrix exponential. We show that while diagonal Padé approximations preserve quadratic stability when going from continuous-time to discrete-time, the converse is not true. We discuss the implications of this result for discretizing switched linear systems. We also show that for continuous-time switched systems which are exponentially stable, but not quadratically stable, a Padé approximation may not preserve stability.

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89–112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge–Kutta methods, Appl. Numer. Math. 58 (2008) 1675–1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed. A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.

Strong Stability Preserving Two-step Runge–Kutta Methods

We investigate the strong stability preserving (SSP) property of two-step Runge–Kutta (TSRK) methods. We prove that all SSP TSRK methods belong to a particularly simple subclass of TSRK methods, in which stages from the previous step are not used. We derive simple order conditions for this subclass. Whereas explicit SSP Runge–Kutta methods have order at most four, we prove that explicit SSP TSRK methods have order at most eight. We present explicit TSRK methods of up to eighth order that were found by numerical search. These methods have larger SSP coefficients than any known methods of the same order of accuracy and may be implemented in a form with relatively modest storage requirements. The usefulness of the TSRK methods is demonstrated through numerical examples, including integration of very high order weighted essentially non-oscillatory discretizations.

Step Sizes for Strong Stability Preservation with Downwind-Biased Operators

Strong stability preserving (SSP) integrators for initial value ODEs preserve temporal monotonicity solution properties in arbitrary norms. All existing SSP methods, including implicit methods, either require small step sizes or achieve only first order accuracy. It is possible to achieve more relaxed step size restrictions in the discretization of hyperbolic PDEs through the use of both upwind- and downwind-biased semidiscretizations. We investigate bounds on the maximum SSP step size for methods that include negative coefficients and downwind-biased semi-discretizations. We prove that the downwind SSP coefficient for linear multistep methods of order greater than one is at most equal to two, while the downwind SSP coefficient for explicit Runge–Kutta methods is at most equal to the number of stages of the method. In contrast, the maximal downwind SSP coefficient for second order Runge–Kutta methods is shown to be unbounded. We present a class of such methods with arbitrarily large SSP coefficient and demonstrate that they achieve second order accuracy for large CFL number.

Boundedness and strong stability of Runge-Kutta methods

In the literature, much attention has been paid to Runge-Kutta methods (RKMs) satisfying special nonlinear stability requirements indicated by the terms total-variation-diminishing (TVD), strong stability preserving (SSP) and monotonicity. Stepsize conditions, guaranteeing these properties, were derived by Shu & Osher (J. Comput. Phys., 77 (1988) pp. 439-471) and in numerous subsequent papers. These special stability requirements imply essential boundedness properties for the numerical methods, among which the property of being total-variation-bounded. Unfortunately, for many RKMs, the above special requirements are violated, so that one cannot conclude in this way that the methods are (total-variation) bounded. In this paper, we study stepsize-conditions for boundedness directly - rather than via the detour of the above special stability properties. We focus on stepsize-conditions which are optimal, in that they are not unnecessarily restrictive. We nd that - in situations where the special stability properties mentioned above are violated - boundedness can be present only within a class of very special RKMs. As a byproduct, our analysis sheds new light on the known theory of monotonicity for RKMs. We obtain separate results for internal and external monotonicity, as well as a new proof of the fundamental relation between monotonicity and Kraaijevanger's coecient. This proof distinguishes itself from older ones in that it is shorter and more transparent, while it requires more simple assumptions on the RKMs under consideration.

On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes

We construct uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for Euler equations of compressible gas dynamics. The same framework also applies to high order accurate finite volume (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO)) schemes. Motivated by Perthame and Shu (1996) [20] and Zhang and Shu (2010) [26], a general framework, for arbitrary order of accuracy, is established to construct a positivity preserving limiter for the finite volume and DG methods with first order Euler forward time discretization solving one-dimensional compressible Euler equations. The limiter can be proven to maintain high order accuracy and is easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the positivity property. Following the idea in Zhang and Shu (2010) [26], we extend this framework to higher dimensions on rectangular meshes in a straightforward way. Numerical tests for the third order DG method are reported to demonstrate the effectiveness of the methods.

Uniform approximation on ideals of multilinear mappings

For each ideal of multilinear mappings $\mathcal{M}$ we explicitly construct a corresponding ideal ${}^{a}\mathcal{M}$ such that multilinear forms in ${}^{a}\mathcal{M}$ are exactly those which can be approximated, in the uniform norm, by multilinear forms in $\mathcal{M}$. This construction is then applied to finite type, compact, weakly compact and absolutely summing multilinear mappings. It is also proved that the correspondence $\mathcal{M} \mapsto {}^{a}\mathcal{M}$ is Aron-Berner stability preserving.

On maximum-principle-satisfying high order schemes for scalar conservation laws

We construct uniformly high order accurate schemes satisfying a strict maximum principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or discontinuous Galerkin (DG) method with first order Euler forward time discretization solving one-dimensional scalar conservation laws. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. It is straightforward to extend the method to two and higher dimensions on rectangular meshes. We also show that the same limiter can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. Numerical tests for both the WENO finite volume scheme and the DG method are reported.

Stepsize Conditions for Boundedness in Numerical Initial Value Problems

For Runge–Kutta methods (RKMs), linear multistep methods (LMMs), and classes of general linear methods (GLMs), much attention has been paid, in the literature, to special nonlinear stability requirements indicated by the terms total-variation-diminishing, strong stability preserving, and monotonicity. Stepsize conditions, guaranteeing these properties, were derived by Shu & Osher [J. Comput. Phys., 77 (1988), pp. 439–471] and in numerous subsequent papers. These special stability requirements imply essential boundedness properties for the numerical methods, among which the property of being total-variation-bounded. Unfortunately, for many well-known methods, the above special requirements are violated, so that one cannot conclude in this way that the methods are (total-variation-)bounded. In this paper, we focus on stepsize conditions for boundedness directly, rather than via the detour of the above special stability properties. We present a generic framework for deriving best possible stepsize conditions which guarantee boundedness of actual RKMs, LMMs, and GLMs, thereby generalizing results on the special stability properties mentioned above.

High Order Strong Stability Preserving Time Discretizations

Strong stability preserving (SSP) high order time discretizations were developed to ensure nonlinear stability properties necessary in the numerical solution of hyperbolic partial differential equa...

Optimal implicit strong stability preserving Runge–Kutta methods

Strong stability preserving (SSP) time discretizations were developed for use with spatial discretizations of partial differential equations that are strongly stable under forward Euler time integration. SSP methods preserve convex boundedness and contractivity properties satisfied by forward Euler, under a modified timestep restriction. We turn to implicit Runge–Kutta methods to alleviate this timestep restriction, and present implicit SSP Runge–Kutta methods which are optimal in the sense that they preserve convex boundedness properties under the largest timestep possible among all methods with a given number of stages and order of accuracy. We consider methods up to order six (the maximal order of SSP Runge–Kutta methods) and up to eleven stages. The numerically optimal methods found are all diagonally implicit, leading us to conjecture that optimal implicit SSP Runge–Kutta methods are diagonally implicit. These methods allow a larger SSP timestep, compared to explicit methods of the same order and number of stages. Numerical tests verify the order and the SSP property of the methods.

Strong stability preserving hybrid methods

This paper is concerned with the strong stability preserving (SSP) time discretizations for semi-discrete systems, obtained from applying the method of lines to time-dependent partial differential equations. We focus on the construction of explicit hybrid methods with nonnegative coefficients, which are a class of multistep methods incorporating a function evaluation at an off-step point. A series of new SSP methods are found. Among them, the low order methods are more efficient than some well known SSP Runge–Kutta or linear multistep methods. In particular, we present some fifth to seventh order methods with nonnegative coefficients, which have healthy CFL coefficients. Finally, some numerical experiments on the Burgers equation are given.

A Numerical Study of Diagonally Split Runge–Kutta Methods for PDEs with Discontinuities

Diagonally split Runge---Kutta (DSRK) time discretization methods are a class of implicit time-stepping schemes which offer both high-order convergence and a form of nonlinear stability known as unconditional contractivity. This combination is not possible within the classes of Runge---Kutta or linear multistep methods and therefore appears promising for the strong stability preserving (SSP) time-stepping community which is generally concerned with computing oscillation-free numerical solutions of PDEs. Using a variety of numerical test problems, we show that although second- and third-order unconditionally contractive DSRK methods do preserve the strong stability property for all time step-sizes, they suffer from order reduction at large step-sizes. Indeed, for time-steps larger than those typically chosen for explicit methods, these DSRK methods behave like first-order implicit methods. This is unfortunate, because it is precisely to allow a large time-step that we choose to use implicit methods. These results suggest that unconditionally contractive DSRK methods are limited in usefulness as they are unable to compete with either the first-order backward Euler method for large step-sizes or with Crank---Nicolson or high-order explicit SSP Runge---Kutta methods for smaller step-sizes. We also present stage order conditions for DSRK methods and show that the observed order reduction is associated with the necessarily low stage order of the unconditionally contractive DSRK methods.

SOLUTION OF 2D SHALLOW WATER EQUATIONS BY GENUINELY MULTIDIMENSIONAL SEMI-DISCRETE CENTRAL SCHEME

A numerical method for solving the two-dimensional shallow water equations was presented. This method was based on the third-order genuinely multidimensional semi-discrete central scheme for spatial discretization and the optimal third-order Strong Stability Preserving (SSP) Runge-Kutta method for time integration. The third-order compact Central Weighted Essentially Non-Oscillatory (CWENO) reconstruction was adopted to guarantee the non-oscillatory behavior of the presented scheme and improve the resolution. Two kinds of source terms were considered in this work. They were evaluated using different approaches. The resulting scheme does not require Riemann solvers or characteristic decomposition, hence it retains all the attractive features of central schemes such as simplicity and high resolution. To evaluate the performance of the presented scheme, several numerical examples were tested. The results demonstrate that our method is efficient, stable and robust.