High Order Explicit Two-Step Runge-Kutta Methods for Parallel Computers

In this paper we study a class of explicit pseudo two-step Runge-Kutta methods (EPTRK methods) with additional weights v. These methods are especially designed for parallel computers. We study s-stage methods with local stage order s and local step order s + 2 and derive a sufficient condition for global convergence order s + 2 for fixed step sizes. Numerical experiments with 4- and 5-stage methods show the influence of this superconvergence condition. However, in general it is not possible to employ the new introduced weights to improve the stability of high order methods. We show, for any given s-stage method with extended weights which fulfills the simplifying conditions B(s) and C(s - 1), the existence of a reduced method with a simple weight vector which has the same linear stability behaviour and the same order

2D bifurcations and Newtonian properties of memristive Chua's circuits

Two interesting properties of Chua's circuits are presented. First, two-parameter bifurcation diagrams of Chua's oscillatory circuits with memristors are presented. To obtain various 2D bifurcation images a substantial numerical effort, possibly with parallel computations, is needed. The numerical algorithm is described first and its numerical code for 2D bifurcation image creation is available for free downloading. Several color 2D images and the corresponding 1D greyscale bifurcation diagrams are included. Secondly, Chua's circuits are linked to Newton's law $\phi ''= F(t,\phi,\phi')/m$ with $\phi=\text{flux}$ , constant m > 0, and the force term $F(t,\phi,\phi')$ containing memory terms. Finally, the jounce scalar equations for Chua's circuits are also discussed

Estimation of a diffusion constant in an equation with a dynamic boundary condition using Fourier analysis

In this paper, we derive a Fourier series expansion of the solution of a onedimensional diffusion equation with a dynamic boundary condition. Due to the boundary condition the spatial operator is not self-adjoint which makes the computation of the Fourier coefficients nonstandard. We use the first eigenfunction, i.e. the first term of the series, to estimate the diffusion constant of urea through a membrane

Splitting methods for the simulation of tumor angiogenesis models

Angiogenesis – the process by which new blood vessels grow into a tissue from surrounding parent vessels – is an important process in many areas of medicine. Here we consider the numerical simulation of a PDE model of tumor-induced angiogenesis. It contains convection (migration), diffusion and reaction terms. Despite the restriction to one specific model, the observations should also be relevant for the solution of similar problems. Our general approach is the method of lines and we use a positivity preserving spatial discretization resulting in a large and in general stiff ODE system. For the solution of this system we consider splitting methods (approximate matrix factorization, Strang-type and source splitting) and Krylov-W-methods. The aim is to reduce the complexity of the implicit relations in the solution process. Advantages and disadvantages of the different approaches are discussed. We compare the methods with respect to efficiency and accuracy of the solution. A Strang-type splitting method combined with approximate matrix factorization is found to be most efficient in the low to modest accuracy range and this range is of interest for the model

Numerical experiments with some explicit pseudo two-step RK methods on a shared memory computer

This paper investigates the performance of two explicit pseudo two-step Runge-Kutta methods of order 5 and 8 for first-order nonstiff ODEs on a parallel shared memory computer. For expensive right hand sides the parallel implementation gives a speedup of 3--4 with respect to the sequential one. Furthermore we compare the codes with the two efficient nonstiff codes DOPRI5 and DOP853. For problems, where the stepsize is determined by accuracy rather than by stability our codes are shown to be more efficient. Key words: Runge-Kutta methods, pseudo two-step Runge-Kutta methods, parallelism. 1 Introduction The arrival of parallel computers influences the development of methods for the numerical solution of a nonstiff initial value problem (IVP) for systems of first-order ordinary differential equations (ODEs) y 0 (t) = f(t; y(t)); y(t 0 ) = y 0 ; y; f 2 R d : (1.1) Although there exist in the literature very efficient sequential numerical methods for solving this problem like multiste..

Numerical experiments with Krylov integrators

We discuss the use of preconditoning in Krylov-W-methods. The preconditioning is based on different operator splitting schemes for reaction-diffusion problems. Comparison of various Krylov codes show that the preconditioned version of ROWMAP works efficient and reliable, especially for large dimensions. Furthermore, parallelism can easily be exploited in the preconditioning. Key words. ROW-methods, stiff initial value problems, Krylov subspaces, preconditioning. AMS(MOS) subject classification. 65L06, 65F10 1 Introduction For the numerical solution of stiff initial value problems y 0 (t) = f(t; y(t)); y(t 0 ) = y 0 2 R n ; (1.1) implicit or linearly implicit methods have to be used due to stability requirements. For large dimensions n these methods spend most of their computing time for the evaluation of the Jacobian and in the solution of the linear equations. Recently, Krylov methods have been used extensively in implicit methods for stiff ordinary differential equations to a..

Construction of highly stable two-step W-methods for ordinary differential equations

We describe the construction of linearly implicit two{step W{methods of high stage order and desirable stability properties for the numerical solution of sti dierential systems. The presented methods are demonstrated to be A{stable and L{stable when the stepsize is held constant. They also preserve good stability properties in a variable stepsize environment under quite demanding conditions imposed on the stepsize pattern