Space-Time Adaptive Fup Multi-Resolution Approach for Boundary-Initial Value Problems
Hrvoje Gotovac; Vedrana Kozuli\'c; and Blaz Gotovac

doi:10.3970/cmc.2010.015.173
Source CMC: Computers, Materials, & Continua, Vol. 15, No. 3, pp. 173-198, 2010
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Keywords Fup basis functions, collocation, partial differential equations, adaptive grid, meshless method, prescribed accuracy of time integration.
Abstract The space-time Adaptive Fup Collocation Method (AFCM) for solving boundary-initial value problems is presented. To solve the one-dimensional initial boundary value problem, we convert the problem into a two-dimensional boundary value problem. This quasi-boundary value problem is then solved simultaneously in the space-time domain with a collocation technique and by using atomic Fup basis functions. The proposed method is a generally meshless methodology because it requires only the addition of collocation points and basis functions over the domain, instead of the classical domain discretization and numerical integration. The grid is adapted progressively by setting the threshold as a direct measure of the solution accuracy at a given resolution level. At higher resolution levels, collocation points are only added in the space-time sub-domains where the solution correction is greater than the prescribed threshold. In contrast to the classical time-stepping schemes, in which globally accumulated errors can arise and which are not easily adapted to multiple time steps, the space-time AFCM covers all space and time multiple scales, while global error is strictly controlled in time by ana priorithreshold.
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