Accuracy analysis of acceleration schemes for stiff multiscale problems

AbstractIn the context of multiscale computations, techniques have recently been developed that enable microscopic simulators to perform macroscopic level tasks (equation-free multiscale computations). The main tool is the so-called coarse-grained time-stepper, which implements an approximation of the unavailable macroscopic time-stepper using only the microscopic simulator. Several schemes were developed to accelerate the coarse-grained time-stepper, exploiting the smoothness in time of the macroscopic dynamics. To date, mainly the stability of these methods was analyzed. In this paper, we focus on their accuracy properties, mainly in the context of parabolic problems. We study the global error of the different methods, compare with explicit stiff ODE solvers, and use the theoretical results to develop more accurate variants. Our theoretical results are confirmed by various numerical experiments

Dynamic refinement for fluid flow simulations with SPH

In this paper, we present a dynamic refinement algorithm for the SPH method where a particle is refined by replacing it with smaller daughter particles. The position of the new particles is calculated by using a square pattern centered at the position of the refined particle. We propose to reduce the error introduced by the refinement by determining the separation of the pattern and the smoothing distance of the daughter particles such that the kernel gradient error is minimized. The results of the simulations using the fully refined domain and the simulations using the dynamic refinement starting from the unrefined domain are compared and are in a good agreement. Better results are obtained when the proposed method to reduce the error is used

Numerical bifurcation analysis of delay differential equations

AbstractNumerical methods for the bifurcation analysis of delay differential equations (DDEs) have only recently received much attention, partially because the theory of DDEs (smoothness, boundedness, stability of solutions) is more complicated and less established than the corresponding theory of ordinary differential equations. As a consequence, no established software packages exist at present for the bifurcation analysis of DDEs. We outline existing numerical methods for the computation and stability analysis of steady-state solutions and periodic solutions of systems of DDEs with several constant delays

An Adaptive Newton--Picard Algorithm with Subspace Iteration for Computing Periodic Solutions

International audienceThis paper is concerned with the efficient computation of periodic orbits in large-scale dynamical systems that arise after spatial discretization of partial differential equations (PDEs). A hybrid Newton–Picard scheme based on the shooting method is derived, which in its simplest form is the recursive projection method (RPM) of Shroff and Keller [SIAM J. Numer. Anal., 30 (1993), pp. 1099–1120] and is used to compute and determine the stability of both stable and unstable periodic orbits. The number of time integrations needed to obtain a solution is shown to be determined only by the system's dynamics. This contrasts with traditional approaches based on Newton's method, for which the number of time integrations grows with the order of the spatial discretiza-tion. Two test examples are given to show the performance of the methods and to illustrate various theoretical points