Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids

Abstract

We propose a simple modification of standard weighted essentially non-oscillatory (WENO) finite volume methods for Cartesian grids, which retains the full spatial order of accuracy of the one-dimensional discretization when applied to nonlinear multidimensional systems of conservation laws. We derive formulas, which allow us to compute high-order accurate point values of the conserved quantities at grid cell interfaces. Using those point values, we can compute a high-order flux at the center of a grid cell interface. Finally, we use those point values to compute high-order accurate averaged fluxes at cell interfaces as needed by a finite volume method. The method is described in detail for the two-dimensional Euler equations of gas dynamics. An extension to the three-dimensional case as well as to other nonlinear systems of conservation laws in divergence form is straightforward. Furthermore, similar ideas can be used to improve the accuracy of WENO type methods for hyperbolic systems which are not in divergence form. Several test computations confirm the high-order accuracy for smooth nonlinear problems.

Appendices

Appendix 1: Explicit Runge–Kutta methods

For the temporal discretization we use explicit Runge–Kutta methods of order 5 and 7, respectively. After discretizing the PDE in space, we obtain a system of ordinary differential equations of the general form

$$\begin{aligned} \frac{d}{dt} Q(t) = {\mathcal L}(Q(t)), \end{aligned}$$

(51)

where $Q(t)$ is a grid function of cell average values of the conserved quantities at time $t$. We discretize the resulting ode system using Runge–Kutta methods of order five and seven. The methods are described by the Butcher tableaus in Tables 10, 11.

Table 10 Butcher tableau of the fifth order accurate Runge–Kutta method from [14]

Table 11 Butcher tableau of a seventh order accurate Runge–Kutta method from [5]

In Fig. 6 we show the stability regions of the two different Runge–Kutta methods used in this paper.

Fig. 6

Regions of absolute stability for RK5 and RK7

Appendix 2: Spatial Reconstruction of Interface Values

In this appendix we give the formulas for the spatial reconstruction of interface averaged values of the conserved quantity used in our implementation of the WENO method. We present the formulas for the reconstruction in the $x$-direction. This is based on a description of WENO methods in [1, 4, 18].

1.1 5th Order Accurate WENO Reconstruction

At grid cell interfaces we compute averaged values of the conserved quantities

$$\begin{aligned} Q_{i\pm \frac{1}{2},j}^{\mp }&= w_1^{\mp } Q_{i\pm \frac{1}{2},j}^{(1\mp )} + w_2^{\mp } Q_{i\pm \frac{1}{2},j}^{(2\mp )} + w_3^{\mp } Q_{i\pm \frac{1}{2},j}^{(3\mp )}, \end{aligned}$$

(52)

with

$$\begin{aligned} Q_{i+\frac{1}{2},j}^{(1-)}&= \frac{1}{3} Q_{i-2,j} - \frac{7}{6} Q_{i-1,j} + \frac{11}{6} Q_{i,j}, \quad Q_{i-\frac{1}{2},j}^{(1+)} = -\frac{1}{6} Q_{i-2,j} + \frac{5}{6} Q_{i-1,j} + \frac{1}{3} Q_{i,j}\nonumber \\ Q_{i+\frac{1}{2},j}^{(2-)}&= -\frac{1}{6} Q_{i-1,j} + \frac{5}{6} Q_{i,j} + \frac{1}{3} Q_{i+1,j}, \quad Q_{i-\frac{1}{2},j}^{(2+)} = \frac{1}{3} Q_{i-1,j} + \frac{5}{6} Q_{i,j} - \frac{1}{6} Q_{i+1,j} \nonumber \\ Q_{i+\frac{1}{2},j}^{(3-)}&= \frac{1}{3} Q_{i,j} + \frac{5}{6} Q_{i+1,j} - \frac{1}{6} Q_{i+2,j}, \quad Q_{i-\frac{1}{2},j}^{(3+)} = \frac{11}{6} Q_{i,j} - \frac{7}{6} Q_{i+1,j} + \frac{1}{3} Q_{i+2,j}.\nonumber \\ \end{aligned}$$

(53)

The coefficients $w_1^\pm ,\ldots , w_3^\pm$ in (52) depend on the local solution structure. In the WENO-Z method suggested by Don and Borges [4], they have the form

$$\begin{aligned} w_j^\pm = \frac{\tilde{w}_j^\pm }{\sum _{i=1}^3 \tilde{w}_i^\pm }, \quad \text{ with } \tilde{w}_j^\pm = \gamma _j^\pm \left( 1 + \left( \frac{\tau _5}{\beta _j + \epsilon } \right) ^p \right) , \end{aligned}$$

(54)

with $p=2$ and $j=1,\ldots ,3$. $\gamma _1^- = \gamma _3^+ = \frac{1}{10},\,\gamma _2^- = \gamma _2^+ = \frac{3}{5},\,\gamma _3^- = \gamma _1^+ = \frac{3}{10},\,\beta _j$ as described in [18, Equation (2.9)], $\tau _5 = |\beta _1-\beta _3|$ and $\epsilon =\varDelta x^4$. The WENO-Z methods are constructed to recover the optimal spatial order of convergence. For other high order WENO methods this may depend stronger on the choice of parameters such as $\epsilon$, see for example [8].

The WENO-JS method is obtained by replacing the computation of $\tilde{w}_j^\pm$ by the formula

$$\begin{aligned} \tilde{w}_j^\pm = \frac{\gamma _j^\pm }{(\epsilon + \beta _j)^2}, \quad j=1,2,3. \end{aligned}$$

(55)

Here the same values are used for $\gamma _j^\pm$ and $\beta _j$, but the parameter $\epsilon$ is replaced by $\epsilon =10^{-6}$.

1.2 7th Order Accurate WENO Reconstruction

Our seventh order spatial reconstruction uses values $Q^{(1 \mp )}, \ldots , Q^{(4\mp )}$ from [1] and computes $Q_{i\pm \frac{1}{2},j}^\mp$ analogously to (52), with weights of the same form

$$\begin{aligned} w_j^\pm = \frac{\tilde{w}_j^\pm }{\sum _{i=1}^4 \tilde{w}_i^\pm }, \quad \text{ with } \tilde{w}_j^\pm = \gamma _j^\pm \left( 1 + \left( \frac{\tau _7}{\beta _j + \epsilon } \right) ^p \right) , \end{aligned}$$

(56)

The $\beta$-terms are set to be equal to $IS_0^4, \ldots , IS_3^4$ as defined in [1, page 415]. For the 7th order WENO-Z method we set $\tau _7 = |\beta _1 + 3 \beta _2 - 3 \beta _3-\beta _4|,\,p=2$ and $\epsilon = \varDelta x^5$, see [4]. For the 7th order WENO-JS method we use

$$\begin{aligned} \tilde{w}_j^\pm = \frac{\gamma _j^\pm }{(\epsilon + \beta _j)^2}, \quad j=1,\ldots ,4. \end{aligned}$$

(57)

with $\epsilon = 10^{-10}$.