Single-Step Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Method for 1-D Euler Equations

Abstract

We propose an explicit, single-step discontinuous Galerkin method on moving grids using the arbitrary Lagrangian–Eulerian approach for one-dimensional Euler equations. The grid is moved with the local fluid velocity modified by some smoothing, which is found to considerably reduce the numerical dissipation introduced by Riemann solvers. The scheme preserves constant states for any mesh motion and we also study its positivity preservation property. Local grid refinement and coarsening are performed to maintain the mesh quality and avoid the appearance of very small or large cells. Second, higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.

Appendices

Appendix A: Numerical Flux

The ALE scheme requires a numerical flux $\hat{\varvec{g}}(\varvec{u}_l,\varvec{u}_r,w)$ which is usually based on some approximate Riemann solver. The numerical flux function is assumed to be consistent in the sense that

$$\begin{aligned} \hat{\varvec{g}}(\varvec{u}, \varvec{u}, w) = \varvec{g}(\varvec{u},w), \qquad \forall\,\varvec{u}\in \mathbb {R}^3, w \in \mathbb {R}.\end{aligned}$$

Since the ALE versions of the numerical fluxes are not so well known, here we list the formulae used in the present work.

1.1 Rusanov Flux

The Rusanov flux is a variant of the Lax–Friedrich flux and is given by

$$\begin{aligned} \hat{\varvec{g}}(\varvec{u}_l,\varvec{u}_r,w) = \frac{1}{2}[ \varvec{g}(\varvec{u}_l,w) + \varvec{g}(\varvec{u}_r,w)] - \frac{1}{2}\lambda _{lr} (\varvec{u}_r - \varvec{u}_l), \end{aligned}$$

where $\lambda _{lr} = \lambda (\varvec{u}_l,\varvec{u}_r,w),$

$$\begin{aligned} \lambda (\varvec{u}_l,\varvec{u}_r,w) = \max \{ |v_l - w|+c_l, |v_r - w|+c_r \}, \end{aligned}$$

which is an estimate of the largest wave speed in the Riemann problem. Since the mesh velocity is close to the fluid velocity, the value of $\lambda$ is close to the local sound speed. Thus, the numerical dissipation is independent of the velocity scale.

1.2 Roe Flux

The Roe scheme [39] is based on a local linearization of the conservation law and then exactly solving the Riemann problem for the linear approximation. The flux can be written as

$$\begin{aligned} \hat{\varvec{g}}(\varvec{u}_l,\varvec{u}_r,w) = \frac{1}{2}[ \varvec{g}(\varvec{u}_l,w) + \varvec{g}(\varvec{u}_r,w)] - \frac{1}{2}|A_w| (\varvec{u}_r - \varvec{u}_l), \end{aligned}$$

where the Roe average matrix $A_w = A_w(\varvec{u}_l,\varvec{u}_r)$ satisfies

$$\begin{aligned} \varvec{g}(\varvec{u}_r,w) - \varvec{g}(\varvec{u}_l,w) = A_w (\varvec{u}_r - \varvec{u}_l), \end{aligned}$$

and we define $|A_w| = R |\varLambda -wI| R^{-1}$. This matrix is evaluated at the Roe average state $\varvec{u}({\bar{\varvec{q}}})$, ${\bar{\varvec{q}}}=\frac{1}{2}(\varvec{q}_l + \varvec{q}_r)$, where $\varvec{q}=\sqrt{\rho }[1, \ v, \ H]^{\text{T}}$ is the parameter vector introduced by Roe.

1.3 HLLC Flux

This is based on a three wave approximate Riemann solver and the particular ALE version we use can also be found in [15]. Define the relative velocity $q= v - w$; then, the numerical flux is given by

$$\begin{aligned} \hat{\varvec{g}}(\varvec{u}_l,\varvec{u}_r,w) = {\left\{ \begin{array}{ll} \varvec{g}(\varvec{u}_l,w), &{} S_l > 0, \\ \varvec{g}^*(\varvec{u}_l^*,w), &{} S_l \le 0< S_M, \\ \varvec{g}^*(\varvec{u}_r^*,w), &{} S_M \le 0 \le S_r, \\ \varvec{g}(\varvec{u}_r,w), &{} S_r < 0, \end{array}\right. } \end{aligned}$$

where the intermediate states are given by

$$\begin{aligned} \varvec{u}_\alpha ^* = \frac{1}{S_\alpha - S_M} \begin{bmatrix} (S_\alpha - q_\alpha ) \rho _\alpha \\ (S_\alpha - q_\alpha )(\rho v)_\alpha + p^* - p_\alpha \\ (S_\alpha - q_\alpha ) E_\alpha - p_\alpha q_\alpha + p^* S_M \end{bmatrix}, \qquad \alpha =l,r, \end{aligned}$$

and

$$\begin{aligned} \varvec{g}^*(\varvec{u},w) = S_M \varvec{u}+ \begin{bmatrix} 0 \\ p^* \\ (S_M + w) p^* \end{bmatrix}, \end{aligned}$$

where

$$\begin{aligned} p^* = \rho _l (q_l - S_l) (q_l - S_M) + p_l = \rho _r (q_r - S_r)(q_r - S_M) + p_r, \end{aligned}$$

which gives $S_M$ as

$$\begin{aligned} S_M = \frac{ \rho _r q_r (S_r - q_r) - \rho _l q_l (S_l - q_l) + p_l - p_r}{\rho _r (S_r - q_r) - \rho _l (S_l - q_l)}. \end{aligned}$$

The signal velocities are defined as

$$\begin{aligned} S_l = \min \{ q_l - c_l, {\hat{v}}-w-{\hat{c}}\}, \qquad S_r = \max \{ q_r + c_r, {\hat{v}}-w+{\hat{c}}\}, \end{aligned}$$

where ${\hat{v}}$, ${\hat{c}}$ are Roe’s average velocity and speed of sound.

Appendix B: Continuous Expansion Runge–Kutta (CERK) Schemes

We use a Runge–Kutta scheme to compute the predicted solution used to compute all the integrals in the DG scheme. In this section, we list down the CERK scheme for the following ODE:

$$\begin{aligned} \frac{\text {d}u}{\text {d}t} = f(u,t). \end{aligned}$$

Given the solution $u^n$ at time $t_n$, the CERK scheme gives a polynomial solution in the time interval $[t_n, t_{n+1})$ of the form

$$\begin{aligned} u(t_n + \theta h) = u^n + h \sum _{s=1}^{n_s} b_s(\theta ) k_s, \qquad \theta \in [0,1], \end{aligned}$$

where $n_s$ is the number of stages and h denotes the time step.

1.1 Second Order (CERK2)

The number of stages is $n_s = 2$ and

$$\begin{aligned} b_1(\theta ) = \theta - \theta ^2/2, \qquad b_2(\theta ) = \theta ^2/2, \end{aligned}$$

and

$$\begin{aligned} k_1 & = f(u^n, t_n), \\ k_2 & = f(u^n + h k_1, t_n + h). \end{aligned}$$

1.2 Third Order (CERK3)

The number of stages is $n_s = 4$ and

$$\begin{aligned} k_1 & = f(u^n, t_n), \\ k_2 & = f(u^n + (12/23) h k_1, t_n + 12 h /23), \\ k_3 & = f(u^n + h ((-68/375) k_1 + (368/375) k_2), t_n + 4h/5),\\ k_4 & = f(u^n + h ((31/144) k_1 + (529/1\,152) k_2 + (125/384) k_3), t_n + h), \end{aligned}$$

and

$$\begin{aligned} b_1(\theta ) & = (41/72) \theta ^3 - (65/48) \theta ^2 + \theta, \\ b_2(\theta ) & = -(529/576) \theta ^3 + (529/384) \theta ^2, \\ b_3(\theta ) & = -(125/192) \theta ^3 + (125/128) \theta ^2, \\ b_4(\theta ) & = \theta ^3 - \theta ^2. \end{aligned}$$