engleski

Space-time adaptive Fup multi-resolution approach for boundary-initial value problems

In this paper we show new space-time AFCM with resolved all space, but also time multiple scales. Essentially, AFCM solves PDE’ s and corresponding boundary conditions in the same way as FCT approximates a function by different resolution levels. All previous existing algorithms with wavelets and splines as well as recent form of the AFCM used localized basis functions only to obtain efficient adaptive strategy, but the PDE is solved by finite difference method on non-uniform adaptive grid (including all levels). In this paper, we present novel form of the AFCM with only Fup basis functions at each level using the collocation framework in space-time domain. Each non-zero level solves only residual of the PDE from all previous levels and gives particular solution correction. Adaptive criterion adds new collocation points in the next level only in the space-time zones where solution correction is greater than the prescribed threshold. Numerical experiments with known analytic solutions show that solution correction on some level is greater than the difference between the analytic solution and overall solution up to that level. It means that an adaptive criterion directly estimates accuracy of the solution even if analytical solution is not a priori known. Moreover, this approach controls the global time integration error and enables a near optimal space-time adaptive grid which includes numerical efficiency and robustness.

boundary-initial value problems; collocation method; atomic basis functions; adaptive criterion

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