HIGH ACCURACY MULTISCALE MULTIGRID COMPUTATION FOR PARTIAL DIFFERENTIAL EQUATIONS

Scientific computing and computer simulation play an increasingly important role in scientific investigation and engineering designs, supplementing traditional experiments, such as in automotive crash studies, global climate change, ocean modeling, medical imaging, and nuclear weapons. The numerical simulation is much cheaper than experimentation for these application areas and it can be used as the third way of science discovery beyond the experimental and theoretical analysis. However, the increasing demand of high resolution solutions of the Partial Differential Equations (PDEs) with less computational time has increased the importance for researchers and engineers to come up with efficient and scalable computational techniques that can solve very large-scale problems. In this dissertation, we build an efficient and highly accurate computational framework to solve PDEs using high order discretization schemes and multiscale multigrid method. Since there is no existing explicit sixth order compact finite difference schemes on a single scale grids, we used Gupta and Zhang’s fourth order compact (FOC) schemes on different scale grids combined with Richardson extrapolation schemes to compute the sixth order solutions on coarse grid. Then we developed an operator based interpolation scheme to approximate the sixth order solutions for every find grid point. We tested our method for 1D/2D/3D Poisson and convection-diffusion equations. We developed a multiscale multigrid method to efficiently solve the linear systems arising from FOC discretizations. It is similar to the full multigrid method, but it does not start from the coarsest level. The major advantage of the multiscale multigrid method is that it has an optimal computational cost similar to that of a full multigrid method and can bring us the converged fourth order solutions on two grids with different scales. In order to keep grid independent convergence for the multiscale multigrid method, line relaxation and plane relaxation are used for 2D and 3D convection diffusion equations with high Reynolds number, respectively. In addition, the residual scaling technique is also applied for high Reynolds number problems. To further optimize the multiscale computation procedure, we developed two new methods. The first method is developed to solve the FOC solutions on two grids using standardW-cycle structure. The novelty of this strategy is that we use the coarse level grid that will be generated in the standard geometric multigrid to solve the discretized equations and achieve higher order accuracy solution. It is more efficient and costs less CPU and memory compared with the V-cycle based multiscale multigrid method. The second method is called the multiple coarse grid computation. It is first proposed in superconvergent multigrid method to speed up the convergence. The basic idea of multigrid superconvergent method is to use multiple coarse grids to generate better correction for the fine grid solution than that from the single coarse grid. However, as far as we know, it has never been used to increase the order of solution accuracy for the fine grid. In this dissertation, we use the idea of multiple coarse grid computation to approximate the fourth order solutions on every coarse grid and fine grid. Then we apply the Richardson extrapolation for every fine grid point to get the sixth order solutions. For parallel implementation, we studied the parallelization and vectorization potential of the Gauss-Seidel relaxation by partitioning the grid space with four colors for solving 3D convection-diffusion equations. We used OpenMP to parallelize the loops in relaxation and residual computation. The numerical results show that the parallelized and the sequential implementation have the same convergence rate and the accuracy of the computed solutions

G-CSC Report 2010

The present report gives a short summary of the research of the Goethe Center for Scientific Computing (G-CSC) of the Goethe University Frankfurt. G-CSC aims at developing and applying methods and tools for modelling and numerical simulation of problems from empirical science and technology. In particular, fast solvers for partial differential equations (i.e. pde) such as robust, parallel, and adaptive multigrid methods and numerical methods for stochastic differential equations are developed. These methods are highly adanvced and allow to solve complex problems.. The G-CSC is organised in departments and interdisciplinary research groups. Departments are localised directly at the G-CSC, while the task of interdisciplinary research groups is to bridge disciplines and to bring scientists form different departments together. Currently, G-CSC consists of the department Simulation and Modelling and the interdisciplinary research group Computational Finance

Schnelle Löser für partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Development of a multiblock solver utilizing the lattice Boltzmann and traditional finite difference methods for fluid flow problems

This dissertation develops the lattice Boltzmann method (LBM) as a strong alternative to traditional numerical methods for solving incompressible fluid flow problems. The LBM outperforms traditional methods on a standalone basis for certain problem cases while for other cases it can be coupled with the traditional methods using domain decomposition. This brings about a composite numerical scheme which associates the efficient numerical attributes of each individual method in the composite scheme with a particular region in the flow domain. Coupled lattice Boltzmann-traditional finite difference procedures are developed and evaluated for CPU time reduction and accuracy of standard test cases. The standard test cases are numerical solutions of the two-dimensional unsteady and steady convection-diffusion equations and two-dimensional steady laminar incompressible flows represented by the backward-facing step flow problem and the flow problem around a cylinder. Multiblock Cartesian grids and hybrid Cartesian-cylindrical grid systems are employed with the composite numerical scheme. A cache-optimized lattice Boltzmann technique is developed to utilize the full computational strength of the LBM. The LBM is an explicit time-marching method and therefore has a time step size limitation. The time step size is limited by the grid spacing and the Mach number. A lattice Boltzmann simulation necessarily requires a low Mach number since it relates to the incompressible Navier-Stokes equations in the low Mach number limit. For steady state problems, the smaller time step results in slow convergence. To improve the time step limitation imposed by the grid spacing, an improved LBM that adopts a new numerical discretization for the advection term has been developed and the results were computed for a convection-diffusion equation and compared with the original LBM. The performance of traditional finite difference methods based on the alternating direction implicit scheme for the convection-diffusion equation and the vorticity-stream function method for the laminar incompressible flow problems is evaluated against the composite numerical scheme. The composite numerical scheme is shown to take lesser CPU time for solving the given benchmark problems

Proceedings of the FEniCS Conference 2017

Proceedings of the FEniCS Conference 2017 that took place 12-14 June 2017 at the University of Luxembourg, Luxembourg

Efficiency and scalability of a two-level Schwarz Algorithm for incompressible and compressible flows

International audienceThis paper studies the application of two-level Schwarz algorithms to several models of Computational Fluid Dynamics. The purpose is to build an algorithm suitable for elliptic and convective models. The sub-domain approximated solution relies on the incomplete lower-upper factorisation (ILU). The algebraic coupling between the coarse grid and the Schwarz preconditioner is discussed. The Deflation Method (DM) and the Balancing Domain Decomposition (BDD) Method are studied for introducing the coarse grid correction as a preconditioner. Standard coarse grids are built with the characteristic or indicator functions of the sub-domains. The building of a set of smooth basis functions (analogous to smoothed-aggregation methods) is considered. A first test problem is the Poisson problem with a discontinuous coefficient. The two options are compared for the standpoint of coarse-grid consistency and for the gain in scalability of the global Schwarz iteration. The advection-diffusion model is then considered as a second test problem. Extensions to compressible flows (together with incompressible flows for comparison) are then proposed. Parallel applications are presented and their performance measured