Abstract
Many phenomena in Physics and Engineering are modeled by Partial Differential Equations. Typically, analytical solutions of such problems are unknown. Hence, numerical schemes are applied for approximating the exact solutions. In order to accurately approximate the exact solutions, very fine grids are required. However, the computations become too costly. In this work, a multiscale iterative approach is utilized for enhancing the accuracy of coarse grid computations which results from a high-order finite difference scheme. The main idea is to run the finite difference computations on a coarse grid until some intermediate time level. Then, the coarse grid results are interpolated and extended to a fine grid using Laplacian Pyramids, a multiscale iterative approach. Finally, the finite difference scheme is employed on the finer grid until a prescribed final time. Comparing the results to those obtained on a fine grid, we see that the convergence rates obtained from the two methods are comparable, while the computational time is significantly reduced. A modified multimodal Laplacian Pyramids algorithm for predicting future values of the solution is also suggested. The method approximates and extends a function based on two or more input modalities coded by a series of multiscale kernels, which are averaged as a convex combination. In this work, the modalities are the numerical model's approximations of the solution of the differential equation and its derivative at previous time steps, and the goal is to predict the solution at a proceeding time step. It can be seen that, by adapting the convex combination of the kernels to local regions of the solution with stronger or weaker gradients, the predicted results are improved.
Original language | English |
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Article number | 115116 |
Journal | Journal of Computational and Applied Mathematics |
Volume | 427 |
DOIs | |
State | Published - 1 Aug 2023 |
Keywords
- Forecasting
- High-order finite difference schemes
- Laplacian Pyramids
All Science Journal Classification (ASJC) codes
- Computational Mathematics
- Applied Mathematics