Search

This paper is concerned with the numerical integration in time of nonlinear Schrödinger equations using different methods preserving the energy or a discrete analogue of it. The Crank–Nicolson method is a well-known method of order |$2$| but is fully implicit and one may prefer a linearly implicit method like the relaxation method introduced in Besse (1998, Analyse numérique des systèmes de Davey-Stewartson. Ph.D. Thesis , Université Bordeaux) for the cubic nonlinear Schrödinger equation. This method is also an energy-preserving method and numerical simulations have shown that its order is |$2$|⁠. In this paper we give a rigorous proof of the order of this relaxation method and propose a generalized version that allows one to deal with general power law nonlinearites. Numerical simulations for different physical models show the efficiency of these methods. [ABSTRACT FROM AUTHOR]

In this paper we mathematically characterize through a Lie formalism the local errors induced by operator splitting when solving nonlinear reaction-diffusion equations, especially in the nonasymptotic regime. The nonasymptotic regime is often attained in practice when the splitting time step is much larger than some of the scales associated with either source terms or the diffusion operator when large gradients are present. In a series of previous works a reduction of the asymptotic orders for a range of large splitting time steps related to very short time scales in the nonlinear source term has been studied, as well as that associated with large gradients but for linearized equations. This study provides a key theoretical step forward since it characterizes the numerical behavior of splitting errors within a more general nonlinear framework, for which new error estimates can be derived by coupling Lie formalism and regularizing effects of the heat equation. The validity of these theoretical results is then assessed in the framework of two numerical applications: a Kolmogorov--Petrovskii--Piskunov-type reaction wave where the influence of stiffness on local error estimates can be thoroughly investigated and a much more complex problem, related to premixed flame dynamics in the low Mach number regime with complex chemistry and detailed transport, for which the present theoretical study provides relevant insights. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a new computational strategy for the simulation of combustion fronts based on adaptive time operator splitting and spatial multiresolution. High-order and dedicated one-step solvers compose the splitting scheme for the reaction, diffusion, and convection subproblems, to independently cope with their inherent numerical difficulties and to properly solve the corresponding temporal scales. Adaptive and thus highly compressed spatial representations for localized fronts originating from multiresolution analysis result in important reductions of memory usage, and hence numerical simulations with sufficiently fine spatial resolution can be performed with standard computational resources. The computational efficiency is further enhanced by splitting time steps established beyond standard stability constraints associated to mesh size or stiff source time scales. The splitting time steps are chosen according to a dynamic splitting technique relying on solid mathematical foundations, which ensures error control of the time integration and successfully discriminates time-varying multi-scale physics. For a given semi-discretized problem, the solution scheme provides dynamic accuracy estimates that reflect the quality of numerical results in terms of numerical errors of integration and compressed spatial representations, for general multi-dimensional problems modeled by stiff PDEs. The strategy is efficiently applied to simulate the propagation of laminar premixed flames interacting with vortex structures, as well as various configurations of self-ignition processes of diffusion flames in similar vortical hydrodynamics fields. A detailed study of the error control is provided and show the potential of the approach. It yields large gains in CPU time, while consistently describing a broad spectrum of space and time scales as well as different physical scenarios. [Copyright &y& Elsevier]

We tackle the numerical simulation of reaction-diffusion equations modeling multi- scale reaction waves. This type of problem induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of steep spatial gradients in the reaction fronts, spatially very localized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. The paper considers a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem. Based on recent theoretical studies of numerical analysis such a strategy leads to a splitting time step which is restricted by neither the fastest scales in the source term nor by stability constraints of the diffusive steps but only by the physics of the phenomenon. We aim thus at solving complete models including all time and space scales within a prescribed accuracy, considering large simulation domains with conventional computing resources. The efficiency is evaluated through the numerical simulation of configurations which were so far out of reach of standard methods in the field of nonlinear chemical dynamics for two-dimensional spiral waves and three-dimensional scroll waves, as an illustration. Future extensions of the proposed strategy to more complex configurations involving other physical phenomena as well as optimization capability on new computer architectures are discussed. [ABSTRACT FROM AUTHOR]

Abstract: We discuss the difficulties of the numerical simulation of a stroke, and we describe the numerical methods which we have developed and used to obtain some realistic results. Nowadays, the computations are performed in two-dimensional slices of a brain, but the strategies to obtain full three-dimensional simulations are explored. This paper is written so as to be understandable by non-mathematicians. [Copyright &y& Elsevier]