Local defect correction for time-dependent problems

The solutions of partial differential equations (PDEs) describing physical phenomena are often characterized by local regions of high activity, i.e., regions where spatial gradients are quite large compared to those in the rest of the domain, where the solution presents a relatively smooth behavior. Examples are encountered in many application areas; one of them is the transport of passive tracers in turbulent flow fields. A passive tracer is a diffusive contaminant in a fluid flow that is present in such low concentration that it does not influence the dynamics of the flow. A few examples of passive tracer transport from everyday life are the exhaust gases from chimneys, smoke from a cigarette, dust particles spread by the wind, etc. Understanding the influence of the main flow on the tracer material is crucial in many engineering applications, such as mixing in chemical reactors or transport of fly ashes in burners. Knowledge of the passive tracer behavior is also of fundamental importance in environmental sciences for studying dispersion of pollutants (e.g. chemical species, radioactive components) in the atmosphere or in the oceans. Since filaments of tracer material are often concentrated only in a very limited part of the computational domain, an efficient numerical solution of this type of problems requires the usage of adaptive grid techniques. In adaptive grid methods, a fine grid spacing and a relatively small time step are adopted only where the relatively large variations occur, so that the computational effort and the memory requirements are minimized. This thesis focuses on a new adaptive grid method for efficiently solving parabolic PDEs characterized by highly localized properties: Local Defect Correction (LDC). In LDC the PDE is integrated on a global uniformcoarse grid and on a local uniform fine grid; the latter is adaptively placed at each time level where the high activity in the solution occurs. At each time step, global and local solution are iteratively combined to ultimately produce a solution on the composite grid, union of global and local grid. In particular, the global approximation provides artificial boundary conditions for the local fine grid problem, while the local approximation is used to estimate the coarse grid local discretization error (or defect) and then to improve the solution globally by means of a defect correction. In the algorithm we propose, the local problem is solved not only with a smaller grid size, but also with a smaller time step than the one used for the global problem. In this way the local solution 124 Summary corrects the error in the global approximation due not only to spatial discretization, but also to temporal discretization. When the LDC iteration has come to a fixed point, it can be proved that the global and the local solution at the new time level coincide at the common points between the two grids. The method described in this thesis extends the LDC technique that was initially introduced in the literature for solving elliptic PDEs. The new LDC algorithm is tested in some concrete examples that illustrate its accuracy, efficiency and robustness. LDC is an iterative process that can be used for practical applications only if it converges sufficiently fast. In this thesis the convergence properties of the LDC method for time-dependent problems are studied in detail, both analytically and by means of numerical experiments. For both one- and two-dimensional problems we investigate the dependency of the LDC convergence rate on the discretization parameters (grid size and time step) for the coarse grid problem. In general, it is observed that LDC converges for any choice of the discretization parameters and that iteration errors are reduced by several orders of magnitude at every iteration. When the coarse and the fine grid problemare discretized applying the finite volume method, special care is needed to guarantee that a discrete conservation property holds for the LDC solution on the composite grid. In fact, if no special precautions are taken, the standard LDC method for time-dependent problems is such that fluxes across the interface between global and local grid are not necessarily in balance. In this thesis we propose a finite volume adapted LDC algorithm for parabolic PDEs. In this algorithm the defect term is adapted in such a way that, at each time step, fluxes across the interface between global and local grid are in balance at convergence of the LDC iteration. The finite volume adapted LDC algorithm is then extended to include a conservative regridding strategy. The strategy guarantees that the composite grid solution satisfies a discrete conservation law also when the local region is moved in time to follow the behavior of the solution. The LDC technique is not restricted to one level of refinement. In this thesis a multilevel LDC method for time-dependent problems is introduced. The time marching strategy is such that time integration at the finer levels can be performed with smaller time steps. Finally, the new, fast converging, conservative and multilevel LDC algorithm is applied to solve a transport problem with highly localized properties. In particular, we test the LDC method on a dipole-wall collision problem. The problem is solved both by LDC and by a Chebyshev-Fourier spectral method. When the two numerical solutions are compared, we see that the two methods yield very similar results. LDC, however, is specifically meant for solving problems whose solutions exhibit local regions of high activity, and for this reason it turns out to be a less complex algorithm than the Chebyshev-Fourier spectral method.