Numerical Solution of Multidimensional Hyperbolic PDEs Using Defect Correction on Adaptive Grids

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• 作者：Mani Razi ; Peter Attar ; Prakash Vedula
• 关键词：Adaptive finite difference solution ; Dimensional splitting ; Modified differential equation ; Multidimensional hyperbolic equations ; Non ; iterative defect correction ; Order of accuracy
• 刊名：Journal of Scientific Computing
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：69
• 期：2
• 页码：581-609
• 全文大小：1,600 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691
• 卷排序：69

文摘

We propose a novel computational approach to obtain high order accurate, finite difference based numerical solutions of hyperbolic partial differential equations, through a combination of grid adaptation, non-iterative defect correction and monotonicity preserving interpolation methods. Reduction of local truncation error is achieved primarily due to a particular choice of an adaptive, non-uniform grid where the local Courant–Friedrich–Levy number is unity, along with non-iterative defect correction. A monotonicity preserving interpolant is further used to map the dependent variables from the non-uniform to uniform grids and vice versa. Dimensional splitting techniques are used to extend the range of application of this method from single to multiple dimensions. Using the monotonicity preserving feature of this interpolant, finite difference schemes with high order of accuracy are developed for solving multidimensional, hyperbolic PDEs. In this work, for the proof of concept, five canonical problems including Liouville equations (in one and two dimensions) with spatially dependent drift coefficients and one-dimensional Burgers equation as well as a two-dimensional nonlinear hyperbolic equation are solved. The results demonstrate four major features of the proposed methodology including: (1) the capability to improve the order of accuracy of difference schemes up to any desired level, (2) the ability to obtain the given level of accuracy at a lower computational cost (or time) when compared to some widely used standard finite difference schemes (3) accurate oscillation-free resolution of discontinuities and (4) the computational simplicity for application to multidimensional problems.