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

Toward transient finite element simulation of thermal deformation of machine tools in real-time

Finite element models without simplifying assumptions can accurately describe the spatial and temporal distribution of heat in machine tools as well as the resulting deformation. In principle, this allows to correct for displacements of the Tool Centre Point and enables high precision manufacturing. However, the computational cost of FE models and restriction to generic algorithms in commercial tools like ANSYS prevents their operational use since simulations have to run faster than real-time. For the case where heat diffusion is slow compared to machine movement, we introduce a tailored implicit–explicit multi-rate time stepping method of higher order based on spectral deferred corrections. Using the open-source FEM library DUNE, we show that fully coupled simulations of the temperature field are possible in real-time for a machine consisting of a stock sliding up and down on rails attached to a stand

Parallel simulation of axon growth in the nervous system

In this paper we discuss a model from neurobiology, which describes theoutgrowth of axons from neurons in the nervous system. The model combines ordinary differential equations, defining the movement of the axons, with parabolic partial differential equations. The parabolic equations model the concentrations of chemicals. The axons are guided by the gradients of these chemoattractant and chemorepellant concentrations.We briefly discuss the numerical techniques that we have used to solve thiscoupled parabolic-gradient system. Special attention is given to the parallel implementation on the SGI Origin 3000 (Teras), a multi-processor machine. For that purpose we use the OpenMP standard. Several parallelization strategies are introduced and tested on the basis of a test example. Simulation results as well as performance results are reported

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

Multirate infinitesimal step methods for atmospheric flow simulation

The numerical solution of the Euler equations requires the treatment of processes in different temporal scales. Sound waves propagate fast compared to advective processes. Based on a spatial discretisation on staggered grids, a multirate time integration procedure is presented here generalising split-explicit Runge-Kutta methods. The advective terms are integrated by a Runge-Kutta method with a macro stepsize restricted by the CFL number. Sound wave terms are treated by small time steps respecting the CFL restriction dictated by the speed of sound. Split-explicit Runge-Kutta methods are generalised by the inclusion of fixed tendencies of previous stages. The stability barrier for the acoustics equation is relaxed by a factor of two. Asymptotic order conditions for the low Mach case are given. The relation to commutator-free exponential integrators is discussed. Stability is analysed for the linear acoustic equation. Numerical tests are executed for the linear acoustics and the nonlinear Euler equations

Generic C++ implementation of high-performance BFS-RBF-based mesh motion schemes

Multi-Dimensional fluid- and structural dynamics problems are solved by computational methods based on Arbitrary Lagrange Euler (ALE) formulation of the continuum mechanical conservation equations. The paper presents a new modification of the radial basis function (RBF) based mesh motion scheme, which combines the RBF interpolation with the breadth-first search (BFS) algorithm. In this emerging domain, it is still unknown which algorithmic approach is the most suitable. Therefore, we realized our C++ implementation on a high abstraction level enabling broad customization and easy extension for further algorithmic research without sacrificing performance