A new numerical strategy with space-time adaptivity and error control for multi-scale streamer discharge simulations

This paper presents a new resolution strategy for multi-scale streamer discharge simulations based on a second order time adaptive integration and space adaptive multiresolution. A classical fluid model is used to describe plasma discharges, considering drift-diffusion equations and the computation of electric field. The proposed numerical method provides a time-space accuracy control of the solution, and thus, an effective accurate resolution independent of the fastest physical time scale. An important improvement of the computational efficiency is achieved whenever the required time steps go beyond standard stability constraints associated with mesh size or source time scales for the resolution of the drift-diffusion equations, whereas the stability constraint related to the dielectric relaxation time scale is respected but with a second order precision. Numerical illustrations show that the strategy can be efficiently applied to simulate the propagation of highly nonlinear ionizing waves as streamer discharges, as well as highly multi-scale nanosecond repetitively pulsed discharges, describing consistently a broad spectrum of space and time scales as well as different physical scenarios for consecutive discharge/post-discharge phases, out of reach of standard non-adaptive methods.Comment: Support of Ecole Centrale Paris is gratefully acknowledged for several month stay of Z. Bonaventura at Laboratory EM2C as visiting Professor. Authors express special thanks to Christian Tenaud (LIMSI-CNRS) for providing the basis of the multiresolution kernel of MR CHORUS, code developed for compressible Navier-Stokes equations (D\'eclaration d'Invention DI 03760-01). Accepted for publication; Journal of Computational Physics (2011) 1-2

ADAPTIVE GRID BASED FINITE DIFFERENCE METHODS FOR SOLUTION OF HYPERBOLIC PDES: APPLICATION TO COMPUTATIONAL MECHANICS AND UNCERTAINTY QUANTIFICATION

Novel finite-difference based numerical methods for solution of linear and nonlinear hyperbolic partial differential equations (PDEs) using adaptive grids are proposed in this dissertation. The overall goal of this research is to improve the accuracy and/or computational efficiency of numerical solutions via the use of adaptive grids and suitable modifications of a given low-order order finite-difference scheme. These methods can be grouped in two broad categories. The first category of adaptive FD methods proposed in the dissertation attempt to reduce the truncation error and/or enhance the accuracy of the underlying numerical schemes via grid distribution alone. Some approaches for grid distribution considered include those based on (i) a moving uniform mesh/domain, (ii) adaptive gradient based refinement (AGBR) and (iii) unit local Courant-Freidrich-Lewy (CFL) number. The improvement in the accuracy which is obtained using these adaptive methods is limited by the underlying scheme formal order of accuracy. In the second category, the CFL based approach proposed in the first category was extended further using defect correction in order to improve the formal order of accuracy and computational efficiency significantly (i.e. by at least one order or higher). The proposed methods in this category are constructed based upon the analysis of the leading order error terms in the modified differential equation associated with the underlying partial differential equation and finite difference scheme. The error terms corresponding to regular and irregular perturbations are identified and the leading order error terms associated with regular perturbations are eliminated using a non-iterative defect correction approach while the error terms associated with irregular perturbations are eliminated using grid adaptation. In the second category of methods involving defect correction (or reduction of leading order terms of truncation error), we explored two different approaches for selection of adaptive grids. These are based on (i) optimal grid dis- tribution and (ii) remapping with monotonicity preserving interpolation. While the first category of methods may be preferred in view of ease of implementation and lower computational complexity, the second category of methods may be preferred in view of greater accuracy and computational efficiency. The two broad categories of methods, which have been applied to problems involving both bounded and unbounded domains, were also extended to multidimensional cases using a dimensional splitting approaches. The performance of these methods was demonstrated using several example problems in computational uncertainty quantification (CUQ) and computational mechanics. The results of the application of the proposed approaches all indicate improvement in both the accuracy and computational efficiency (by about three orders of magnitude in some selected cases) of underlying schemes. In the context of CUQ, all three proposed adaptive finite different solvers are combined with the Gauss-quadrature sampling technique in excitation space to obtain statistical quantities of interest for dynamical systems with parametric uncertainties from the solution of Liouville equation, which is a linear hyperbolic PDE. The numerical results for four canonical UQ problems show both enhanced computational efficiency and improved accuracy of the proposed adaptive FD solution of the Liouville equation compared to its standard/fixed domain FD solutions. Moreover, the results for canonical test problems in computational mechanics indicate that the proposed approach for increasing the formal order of the underlying FD scheme can be easily implemented in multidimensional spaces and gives an oscillation-free numerical solution with a desired order of accuracy in a reasonable computational time. This approach is shown to provide a better computational time compared to both the underlying scheme (by about three orders of magnitude) and standard FD methods of the same order of accuracy