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

Abstract

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.

Appendix: Defect Correction for the MacCormack Scheme

The Truncation error analysis of the MacCormack equation (Eq. 20) using the method of modified equation [73] followed by a heuristic numerical stability analysis [33] gives $D_i\Delta t-\Delta x_{i-\frac{1}{2}}=0$ as the characteristic equation for the singular perturbations to the modified differential equation. Therefore in order to obtain the expression for non-iterative defect correction (DC), the spatial increments $\Delta x_{i-\frac{1}{2}}$ and $\Delta x_{i+\frac{1}{2}}$ should be considered spatially dependent variables. Using method of modified equation and after a considerable algebra, this expression is obtained as follows:

$$\begin{aligned} \text {DC}= & {} \frac{\Delta t}{2}\left( D_{i+1}-D_i\right) \left( \frac{\partial ^2(Du)}{\partial x^2}\right) _i+\Delta t^2\bigg (u_i\bigg (\frac{(D_i^{\prime })^3}{6} \nonumber \\&+\, \frac{1}{4}\left( \frac{dD}{dx}\right) _{i+1}\left( D\frac{d^2D}{dx^2}\right) _i+D_{i+1}\left( \frac{dD}{dx}\frac{d^2D}{dx^2}\right) _i\nonumber \\&-\, \frac{1}{12}\left( D\frac{dD}{dx}\frac{d^2D}{dx^2}\right) _i-\frac{1}{6}\left( D\right) _{i+1}^2\left( \frac{d^3D}{dx^3}\right) _{i}\nonumber \\&+\, \frac{1}{2}D_{i+1}\left( D\frac{d^3D}{dx^3}\right) _i+\frac{1}{3}\left( D^2\frac{d^3D}{dx^3}\right) _i\bigg ) \nonumber \\&+\, \frac{\partial u}{\partial x}\bigg (\frac{1}{2}\left( \frac{dD}{dx}\right) _{i+1}\left( D\frac{dD}{dx}\right) _i+\frac{3}{2}D_{i+1}\left( \frac{dD}{dx}\right) _i^2\nonumber \\&-\, \frac{1}{3}\left( D\left( \frac{dD}{dx}\right) ^2\right) _i-\frac{1}{2}D_{i+1}^2\left( \frac{d^2D}{dx^2}\right) _i+\frac{7}{4}D_{i+1}\left( D\frac{d^2D}{dx^2}\right) _i\nonumber \\&-\, \frac{5}{6}\left( D^2\frac{d^2D}{dx^2}\right) _i\bigg )+ \frac{\partial ^2 u}{\partial x^2}\bigg (\frac{1}{4}D_i^2\left( \frac{dD}{dx}\right) _{i+1}\nonumber \\&-\, \frac{1}{2}D_{i+1}^2\left( \frac{dD}{dx}\right) _{i}+\frac{9}{4}D_{i+1}\left( D\frac{dD}{dx}\right) _{i} \nonumber \\&-\, \frac{5}{4}\left( D^2\frac{dD}{dx}\right) _{i}\bigg )+\frac{\partial ^3 u}{\partial x^3}\bigg (-\frac{1}{6}D_i\left( D\right) _{i+1}^2+\frac{1}{2}D_{i+1}\left( D\right) _{i}-\frac{1}{3}D_i^3 \bigg ) \nonumber \\&+\, {\mathcal {O}}\left( \Delta x_{i-\frac{1}{2}}^3+\Delta x_{i+\frac{1}{2}}^3+\Delta t^3\right) . \end{aligned}$$

(43)