This thesis investigates pseudo time stepping as an implicit solver for non-uniform meshes. Specifically, on meshes with small cells. Regions with small cells can be treated with a time stepping scheme known as mixed explicit implicit time stepping, which treats small cells implicitly to ensure stability. The goal of this thesis is to examine whether pseudo time stepping is a good solver for the implicit part. To investigate this, results from pseudo time stepping on a uniform mesh are first compared with a direct implicit solver. This is done to ensure that both methods produce the same results. Then, tests are conducted on a non-uniform mesh, featuring one small cell in the center of the domain. The aim of these tests is to evaluate how the parameters in pseudo time stepping contribute to the accuracy and number of iterations in a small mesh. With respect to this, the thesis aims to answer whether and when pseudo time stepping is a feasible solver. The results show promising potential in pseudo time stepping, as similar results to the direct solver on the uniform mesh are achieved with few iterations. It also shows feasible iteration numbers for the non-equidistant case, nearly regardless of number of grid points or the size of the small cell.

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