Ole Møller Nielsen -
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I am investigating the potential for using these desirable properties of wavelets for solving partial differential equations. One approach is to work with an adaptive grid; The grid density must then be high where the solution changes rapidly while it can be low where the solution is smooth. The wavelets are then used to detect where steep gradients (for example shocks) are forming. The picture below shows a numerical solution to Burgers' equation. The circles on the x-axis indicate grid points as chosen by the wavelet method. It is seen that the grid density is high where the shock is forming. Click here or on the picture to see an animation of this process.