In the present investigation, a space-time adaptive multiresolution method is developed to solve evolutionary PDEs, typically encountered in fluid mechanics. The new method is based on a multiresolution analysis which allows to reduce the number of active grid points significantly by refining the grid automatically in regions of steep gradients, while in regions where the solution is smooth coarse grids are used. The method is applied to the one-dimensional Burgers equation as a classical example of nonlinear advection-diffusion problems and then extended to the incompressible two-dimensional Navier-Stokes equations. To study the near wall behavior of two-dimensional vortical flows a recently revived, dipole collision with a straight wall is considered as a benchmark. After that an extension to interactions with curved walls of concave or convex shape is done using the volume penalization method. The space discretization is based on a second order central finite difference method with symmetric stencil over an adaptive grid. The grid adaptation strategy exploits the local regularity of the solution estimated via the wavelet coefficients at a given time step. Nonlinear thresholding of the wavelet coefficients in a one-to-one correspondence with the grid allows to reduce the number of grid points significantly. Then the grid for the next time step is extended by adding a safety zone in wavelet coefficient space around the retained coefficients in space and scale. With the use of Harten's point value multiresolution framework, general boundary conditions can be applied to the equations. For time integration explicit Runge-Kutta methods of different order are implemented, either with fixed or adaptive time stepping. The obtained results show that the CPU time of the adaptive simulations can be significantly reduced with respect to simulations on a regular grid. Nevertheless the accuracy order of the underlying numerical scheme is preserved.
Development of an adaptive multi-resolution method to study the near wall behavior of two-dimensional vortical flows