Teo Roldán

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.

Starting algorithms for some DIRK methods

When differential 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 construct starting algorithms for some DIRK methods. Numerical experiments have been carried out in order to compare the initializers studied in this paper with others used in the literature.

Design and Implementation of Predictors for Additive Semi-Implicit Runge-Kutta Methods

Space discretization of some time-dependent partial differential equations gives rise to stiff systems of ordinary differential equations. In this case, implicit methods should be used, and therefore, in general, nonlinear systems must be solved. The solutions to these systems are approximated by iterative schemes, and, in order to obtain an efficient code, good initializers should be used. Recently, a parallel code based on some Runge-Kutta and additive Runge-Kutta methods has been constructed, focusing especially on additive semi-implicit Runge-Kutta (ASIRK) methods. The aim of the present paper is to develop efficient initializers for these methods.

Construction of Additive Semi-Implicit Runge---Kutta Methods with Low-Storage Requirements

The final publication is available at Springer via http://dx.doi.org/ 10.1007/s10915-015-0116-2Space discretization of some time-dependent partial differential equations gives rise to systems of ordinary differential equations in additive form whose terms have different stiffness properties. In these cases, implicit methods should be used to integrate the stiff terms while efficient explicit methods can be used for the non-stiff part of the problem. However, for systems with a large number of equations, memory storage requirement is also an important issue. When the high dimension of the problem compromises the computer memory capacity, it is important to incorporate low memory usage to some other properties of the scheme. In this paper we consider Additive Semi-Implicit Runge–Kutta (ASIRK) methods, a class of implicit-explicit Runge–Kutta methods for additive differential systems. We construct two second order 3-stage ASIRK schemes with low-storage requirements. Having in mind problems with stiffness parameters, besides accuracy and stability properties, we also impose stiff accuracy conditions. The numerical experiments done show the advantages of the new methods.

Starting algorithms for low stage order RKN methods

When second order differential equations are solved with Runge-Kutta-Nystrom methods, the computational effort is dominated by the cost of solving the nonlinear system. That is why it is important to have good starting values to begin the iterations. In this paper we consider a type of starting algorithms without additional computational cost. We study the general order conditions and the maximum order achieved when the Runge-Kutta-Nystrom method satisfies some simplifying assumptions.

Order Barrier for Low-Storage DIRK Methods with Positive Weights

In this paper we study an order barrier for low-storage diagonally implicit Runge–Kutta (DIRK) methods with positive weights. The Butcher matrix for these schemes, that can be implemented with only two memory registers in the van der Houwen implementation, has a special structure that restricts the number of free parameters of the method. We prove that third order low-storage DIRK methods must contain negative weights, obtaining the order barrier

Positivity-preserving and entropy-decaying IMEX methods

p\le 2

Numerical Simulation in Physics and Engineering

p≤2 for these schemes. This result extends the well known one for symplectic DIRK methods, which are a particular case of low-storage DIRK methods. Some other properties of second order low-storage DIRK methods are given.