Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction-diffusion systems

We present a fully adaptive multiresolution scheme for spatially two-dimensional, possibly degenerate reaction-diffusion systems, focusing on combustion models and models of pattern formation and chemotaxis in mathematical biology. Solutions of these equations in these applications exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. The multiresolution scheme is based on finite volume discretizations with explicit time stepping. The multiresolution representation of the solution is stored in a graded tree. By a thresholding procedure, namely the elimination of leaves that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen optimally, in the sense that the total error of the adaptive scheme is of the same slope as that of the reference finite volume scheme. Since chemical reactions involve a large range of temporal scales, but are spatially well localized (especially in the combustion model), a locally varying adaptive time stepping strategy is applied. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled.Comment: 27 pages, 14 figure

A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology

This work deals with the numerical solution of the monodomain and bidomain models of electrical activity of myocardial tissue. The bidomain model is a system consisting of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the so-called gating variable. In the simpler sub-case of the monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple models for the membrane and ionic currents are considered, the Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates thatthese methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, an optimalthreshold for discarding non-significant information in the multiresolution representation of the solution is derived, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms.Comment: 25 pages, 41 figure

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

An adaptive multiresolution scheme with local time stepping for evolutionary PDEs

We present a fully adaptive numerical scheme for evolutionary PDEs in Cartesian geometry based on a second-order finite volume discretization. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. For time discretization we use an explicit Runge-Kutta scheme of second-order with a scale-dependent time step. On the finest scale the size of the time step is imposed by the stability condition of the explicit scheme. On larger scales, the time step can be increased without violating the stability requirement of the explicit scheme. The implementation uses a dynamic tree data structure. Numerical validations for test problems in one space dimension demonstrate the efficiency and accuracy of the local time-stepping scheme with respect to both multiresolution scheme with global time stepping and finite volume scheme on a regular grid. Fully adaptive three-dimensional computations for reaction-diffusion equations illustrate the additional speed-up of the local time stepping for a thermo-diffusive flame instability. (C) 2007 Elsevier Inc. All rights reserved.22783758378

Quench simulation of superconducting magnets with commercial multi-physics software

The simulation of quenches in superconducting magnets is a multiphysics problem of highest complexity. Operated at 1.9 K above absolute zero, the material properties of superconductors and superfluid helium vary by several orders of magnitude over a range of only 10 K. The heat transfer from metal to helium goes through different transfer and boiling regimes as a function of temperature, heat flux, and transferred energy. Electrical, magnetic, thermal, and fluid dynamic effects are intimately coupled, yet live on vastly different time and spatial scales. While the physical models may be the same in all cases, it is an open debate whether the user should opt for commercial multiphysics software like ANSYS or COMSOL, write customized models based on general purpose network solvers like SPICE, or implement the physics models and numerical solvers entirely in custom software like the QP3, THEA, and ROXIE codes currently in use at the European Organisation for Nuclear Research (CERN). Each approach has its strengths and limitations, some related to performance, others to usability and maintainability, and others again to the flexibility of material parameterizations. In this context the master thesis mainly involves the study of the strengths and limitations of the first approach. The primary goal of the thesis is to build a 1D numerical model representing a superconducting wire based on existing physical models. An adiabatic model has been constructed, to solve one of the five boundary value problems involved in the quench, both in ANSYS and in COMSOL. The temperature dependent material properties and loads are defined using function tools in COMSOL and by creating look up tables in ANSYS. The models were validated with QP3 and compared in terms of performance, stability and accuracy. The helium-cooled model is built only in ANSYS. The model solves two of the five boundary value problems simultaneously as a coupled problem. Apart from generic numerical code (transient thermal analysis), a separate algorithm is needed to define the non-linear heat transfer between the metal and the helium. For this ANSYS Parametric Design Language (APDL) scripts are used. During the analysis the ANSYS transient thermal codes are executed several times within a loop. There are three different types of helium cooled models. All models were validated with QP3. The results obtained from comparisons show that the adiabatic models were able to simulate quenches with the desired accuracy. The adiabatic analysis in the commercial simulation tools is more efficient and stable for various scale of spatial discretization. Similarly, the helium-cooled models are able to simulate quenches with satisfactory accuracy. Nevertheless, the models are not compatible with automatic time stepping method of the simulation environment. The use of fixed time stepping method in the models resulted the coupled analysis in ANSYS to be far more time consuming than in QP3

쌍곡 보존 법칙들을 풀기 위한 고차정확도 수치기법에 대한 연구

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2017. 2. 강명주.In this thesis, we develop efficient and high order accurate numerical schemes for solving hyperbolic conservation laws such as the Euler equation and the ideal MHD(Magnetohydrodynamics) equations. The first scheme we propose is the \textit{wavelet-based adaptive WENO method}. The Finite difference WENO scheme is one of the popular numerical schemes for application to hyperbolic conservation laws. The scheme has high order accuracy, robustness and stable property. On the other hand, the WENO scheme is computationally expensive since it performs characteristic decomposition and computes non-linear weights for WENO interpolations. In order to overcome the drawback, we propose the adaptation technique that applies WENO differentiation for only discontinuous regions and central differentiation without characteristic decomposition for the other regions. Therefore continuous and discontinuous regions should be appropriately classified so that the adaptation method successfully works. In the wavelet-based WENO method, singularities are detected by analyzing wavelet coefficients. Such coefficients are also used to reconstruct the compressed informations. Secondly, we propose \textit{central-upwind schemes with modified MLP(multi-dimensional limiting process)}. This scheme decreases computational cost by simplifying the scheme itself, while the first method achieve efficiency by skipping grid points. Generally the high-order central difference schemes for conservation laws have no Riemann solvers and characteristic decompositions but tend to smear linear discontinuities. To overcome the drawback of central-upwind schemes, we use the multi-dimensional limiting process which utilizes multi-dimensional information for slope limitation to control the oscillations across discontinuities for multi-dimensional applications.1 Introduction 1 2 Governing Equations 7 2.1 Hyperbolic Conservation Laws 7 2.2 Euler equation 9 2.2.1 Model equation 9 2.2.2 Eigen-structure 10 2.3 Ideal MHD equation 14 2.3.1 Model equation 14 2.3.2 Eigen-Structure 15 2.4 The r B = 0 Constraint in MHD Codes 20 2.4.1 Constraints Transport Method 20 2.4.2 Divergence cleaning technique 23 3 Wavelet-based Adaptation Strategy with Finite Dierence WENO scheme 28 3.1 Finite Dierence WENO scheme 28 3.1.1 Characteristic Decomposition 28 3.1.2 WENO-type Approximations 30 3.2 Wavelet Analysis 32 3.2.1 Multi-resolution Approximations 32 3.2.2 Orthogonal Wavelets 36 3.2.3 Constructing Wavelets 37 3.2.4 Biorthogonal Wavelets 38 3.2.5 Interpolating Scaling Function 40 3.3 Adaptive wavelet Collocation Method 45 3.3.1 Interpolating Wavelets 47 3.3.2 Lifting Scheme 52 3.3.3 Lifting Donoho wavelets family 56 3.3.4 The Lifted interpolating wavelet transform 58 3.3.5 Compression 64 3.4 Wavelet-based Adaptive WENO scheme 65 3.4.1 Adjacent Zone 65 3.4.2 Methodology for Spatial discretizations 66 3.4.3 Time Integration 67 3.4.4 Conservation error and boundary treatment 68 3.4.5 Overall Process 69 3.5 Numerical results 69 3.5.1 1-dimensional equations 70 3.5.2 2-dimensional Euler equations 71 3.5.3 2-dimensional MHD equations 83 4 Combination of Central-Upwind Method and Multi-dimensional Limiting Process 90 4.1 Review of Central-Upwind method 92 4.2 Review of Multi-dimensional Limiting Process 95 4.3 Central-Upwind method with Modied MLP limiter 98 4.4 Numerical results 104 4.4.1 Linear advection equation 105 4.4.2 Burger's equation 106 4.4.3 2D Euler system - Four shocks 106 4.4.4 2D Euler system - Rayleigh-Taylor instability 107 4.4.5 2D Euler system - Double Mach reection of a strong shock 109 5 Conclusions 111 Abstract (in Korean) 121Docto