Inmaculada Higueras

On Strong Stability Preserving Time Discretization Methods

Over the last few years, great effort has been made to develop high order strong stability preserving (SSP) Runge–Kutta methods. These methods have a nonlinear stability property that makes them suitable for the time integration of ODEs that arise from a method of lines approximation of hyperbolic conservation laws. Basically, this stability property is a monotonicity property for the internal stages and the numerical solution. Recently Ferracina and Spijker have established a link between stepsize restrictions for monotonicity and the already known stepsize restrictions for contractivity. Hence the extensive research on contractivity can be transferred to the SSP context. In this paper we consider monotonicity issues for arbitrary norms and linear and nonlinear problems. We collect and review some known results and relate them with the ones obtained in the SSP context.

Stability preserving integration of index-1 DAEs

For index-1 DAEs with properly stated leading term, we characterize dissipative and contractive flows and study how the qualitative properties of the DAE solutions are reflected by numerical approximations. The best situation occurs when the discretization and the decoupling procedure commute. It turns out that this is the case if the relevant part of the inherent regular ODE has a constant state space. Different kinds of reformulations are studied to obtain this property. Those reformulations might be expensive, hence, in order to avoid them, criteria ensuring the given DAE to be numerically equivalent to a numerically qualified representation are proved.

Logarithmic Norms for Matrix Pencils

We extend the usual concepts of least upper bound norm and logarithmic norm of a matrix to matrix pencils. Properties of these seminorms and logarithmic norms are derived. This logarithmic norm can be used to study the growth of the solutions of linear variable coefficient differential algebraic systems.

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.

Monotonicity for Runge-Kutta Methods: Inner Product Norms

An important class of ordinary differential systems is that whose solutions satisfy a monotonicity property for a given norm. For these problems, a natural requirement for the numerical solution is the reflection of this monotonicity property, perhaps under certain stepsize restriction. For Runge–Kutta methods, when the applied norm is an arbitrary one, the stepsize restrictions depend on the radius of absolute monotonicity. However for many problems, monotonicity holds for inner product norms and therefore it makes sense to restrict the analysis to this class of norms to obtain, if possible, less restrictive results. In this paper, we consider monotonicity issues for Runge–Kutta methods when the applied norm is an inner product norm.

IRK methods for DAE: starting algorithms

Abstract When semi-explicit differential-algebraic equations are solved with implicit Runge–Kutta methods, the computational effort is dominated by the cost of solving the nonlinear systems. That is why it is important to have good starting values to begin the iterations. In this paper we study a type of starting algorithms, without additional computational cost, in the case of index-1 DAE. The order of the starting values is defined, and by using DA-series and rooted trees we obtain their general order conditions. If the RK satisfies some simplified assumptions, then the maximum order can be obtained.

On stability issues for IMEX schemes applied to 1D scalar hyperbolic equations with stiff reaction terms

The application of a Method of Lines to a hyperbolic PDE with source terms gives rise to a system of ODEs containing terms that may have very different stiffness properties. In this case, Implicit-Explicit Runge-Kutta (IMEX-RK) schemes are particularly useful as high order time integrators because they allow an explicit handling of the convective terms, which can be discretized using the highly developed shock capturing technology, together with an implicit treatment of the source terms, necessary for stability reasons. Motivated by the structure of the source term in a model problem introduced by LeVeque and Yee in [J. Comput. Phys. 86 (1990)], in this paper we study the preservation of certain invariant regions as a weak stability constraint. For the class of source terms considered in this paper, the unit interval is an invariant region for the model balance law. In the first part of the paper, we consider first order time discretizations, which are the basic building blocks of higher order IMEX-RK schemes, and study the conditions that guarantee that [0, 1] is also an invariant region for the numerical scheme. In the second part of the paper, we study the conditions that ensure the preservation of this property for higher order IMEX schemes.

Characterizing Strong Stability Preserving Additive Runge-Kutta Methods

Space discretization of some time dependent partial differential equations give rise to ordinary differential equations containing additive terms with different stiffness properties. In these situations, additive Runge-Kutta (additive RK) methods are used.For additive RK methods the curve of absolute monotonicity gives stepsize restrictions for monotonicity. Necessary conditions for nontrivial curves of absolute monotonicity are the nonnegativity of the additive RK coefficients and some inequalities on some incidence matrices. In this paper we characterize strong stability preserving additive Runge-Kutta methods giving some order barriers and structural properties.

How Close Can the Logarithmic Norm of a Matrix Pencil Come to the Spectral Abscissa

Given a least upper bound norm, the usefulness of the concept of logarithmic norm depends on how closely the logarithmic norm approximates the spectral abscissa. To study this problem, Strom introduced in 1975 the concepts of logarithmically optimal norm and

Logarithmic Norms and Nonlinear DAE Stability

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