Abstract
Adaptive wavelet collocation methods use wavelet transform and filtering to generate adaptive grids. However, if the boundary moves, the grid becomes aberrant. It baffles wavelet transform and makes the adaptive wavelet methods lose advantages on computational efficiency. This paper develops a series of methods or skills to effectively perform wavelet transform and to generate adaptive grids for one-dimensional movable boundary problems. The methods remain the original inner grid points and keep the grid in the original-nested structure, in order to remain scanty during the whole computing process. For boundary extending, the adaptive wavelet program begins to run on the very new grid beyond the original boundary once it reaches a nested structure, which is called the Intermittent Adaptive Method as a consequence. If the boundary extends tremendously, the new nested grids can be combined to a greater nested grid for further efficiency, which is named the Grid Combine Method. While for boundary contracting, a fictitious boundary is addressed to replace the original boundary that will recede, so wavelet transform can be successfully performed on the original nested grid. Finally, two numerical tests, local features moving and gas gun, were resolved and discussed to show the evolution process of the adaptive grids with the boundaries moving. For boundary contracting, the valid points decrease because the computation domain recedes; while for boundary extending, the valid point numbers vary between a range that almost remains unchanged.
Highlights
If the boundary moves, the grid becomes aberrant. It baffles wavelet transform and makes the adaptive wavelet methods lose advantages on computational efficiency. is paper develops a series of methods or skills to effectively perform wavelet transform and to generate adaptive grids for one-dimensional movable boundary problems. e methods remain the original inner grid points and keep the grid in the original-nested structure, in order to remain scanty during the whole computing process
Because any wavelet and scale function maps a collocation, so the irregular grid is generated by deleting the small-coefficient wavelets. ere are two skills that should be noted: (1) for evolution equations, some collocations near the rough regions should be remained to capture the evolution of local features of the time step, their coefficients are truly smaller than the threshold parameter ε; (2) wavelet decomposition should be performed on the irregular grid to decrease computational costs which may become equal to iteration
E Euler equations (11) are resolved by 1st order upwind scheme [17] based on the Steger–Warming flux vector split method on adaptive grids. e ordinary differential equation (13) is resolved by the explicit Euler method
Summary
Ere are two skills that should be noted: (1) for evolution equations, some collocations near the rough regions should be remained to capture the evolution of local features of the time step, their coefficients are truly smaller than the threshold parameter ε; (2) wavelet decomposition should be performed on the irregular grid to decrease computational costs which may become equal to iteration. E neighbor points in the neighbor domain of the notable are “deleted” firstly after wavelet filtering, but “added” after checking the valid, which are used to predict the local structures and transit the adaptive grid from the current to the time step Both the notable and the neighbor are included in. It should be borne in mind that a main purpose of developing adaptive wavelet methods is to save computational costs
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