A High-Order Maximum-Principle-Satisfying Discontinuous Galerkin Method for the Level Set Problem

Abstract

Level set (LS) method is a widely used interface capturing method. In the simulations of incompressible two-phase flows, in order to avoid discontinuities at interfaces, the LS function is usually taken as a smeared-out Heaviside function bounded on [0, 1] and advected by a given velocity field $\mathbf {u}$ obtained from the solution of the incompressible Navier-Stokes equations. In the incompressible limit $\nabla \cdot \mathbf {u}=0$, the advection equation for the LS function can be written and discretized in conservative form. However, due to numerical errors, the resulting velocity field is in general not divergence free which leads to the solution of the advection equation in conservative form does not satisfy the maximum principle. To overcome this issue, in this work, we develop a high-order discontinuous Galerkin (DG) method to directly solve the advection equation for the LS function in non-conservative form. Moreover, we prove that by applying a linear scaling limiter, the proposed method together with a strong stability preserving (SSP) time discretization scheme can satisfy the strict maximum principle under a suitable CFL condition. Numerical simulations of several well-known benchmark problems, including the application to incompressible two-phase flows, are presented to demonstrate the high-order accuracy and maximum-principle-satisfying property of the proposed method.

A Proof of Theorem 2

Consider an arbitrary order $P^k$ approximation ($k\ge 1$). By using an N-point Gauss-Lobbato and an L-point Gauss-Legendre quadrature rules exact for single-variable polynomials of degree k, we can represent the solution $\varphi _h(x,y)$ along the line $y=y_j^\alpha$ ($1\le \alpha \le L$) as

$$\begin{aligned} \varphi _h(x,y_j^\alpha )=\sum \limits _{\beta =1}^N\varphi _{{\widehat{\beta }},\alpha }\prod \limits _{\gamma =1,\gamma \ne \beta }^N \frac{(x-\widehat{x}_i^\gamma )}{(\widehat{x}_i^\beta -\widehat{x}_i^\gamma )}\overset{def}{=}\sum \limits _{\beta =1}^N\varphi _{{\widehat{\beta }},\alpha }h^{(\beta )}(x). \end{aligned}$$

(69)

Here, $\varphi _{{\widehat{\beta }},\alpha }\overset{def}{=}\varphi _h(\widehat{x}_i^\beta ,y_j^\alpha )$. Taking derivative once with respect to x results in

$$\begin{aligned} \partial _x\varphi _h(x,y_j^\alpha )=\sum \limits _{\beta =1}^N\varphi _{{\widehat{\beta }},\alpha } \frac{\sum \limits _{\begin{array}{c} \kappa =1\\ \kappa \ne \beta \end{array}}^N\prod \limits _{\begin{array}{c} s=1\\ s\ne \beta ,\kappa \end{array}}^{N}(x-\widehat{x}_i^s)}{\prod \limits _{\begin{array}{c} \gamma =1\\ \gamma \ne \beta \end{array}}^N(\widehat{x}_i^\beta -\widehat{x}_i^\gamma )}\overset{def}{=}\sum \limits _{\beta =1}^N\varphi _{{\widehat{\beta }},\alpha }h_x^{(\beta )}(x). \end{aligned}$$

(70)

Similarly, we have

$$\begin{aligned} \partial _y\varphi _h(x_i^\alpha ,y)=\sum \limits _{\beta =1}^N\varphi _{\alpha ,{\widehat{\beta }}} \frac{\sum \limits _{\begin{array}{c} \kappa =1\\ \kappa \ne \beta \end{array}}^N\prod \limits _{\begin{array}{c} s=1\\ s\ne \beta ,\kappa \end{array}}^{N}(y-\widehat{y}_j^s)}{\prod \limits _{\begin{array}{c} \gamma =1\\ \gamma \ne \beta \end{array}}^N(\widehat{y}_j^\beta -\widehat{y}_j^\gamma )}\overset{def}{=}\sum \limits _{\beta =1}^N\varphi _{\alpha ,{\widehat{\beta }}}h_y^{(\beta )}(y). \end{aligned}$$

(71)

Here, $\varphi _{\alpha ,{\widehat{\beta }}}\overset{def}{=}\varphi _h(x_i^\alpha , \widehat{y}_j^\beta )$. Then, the surface integral in Eq. (27) can be further written as

$$\begin{aligned} \begin{aligned}&\frac{\varDelta t}{|I_{i,j}|}\int _{I_{i,j}}\mathbf {u}_h\cdot \nabla \varphi _hd\mathbf {x}\\&\quad =\lambda _x\sum \limits _{\alpha =1}^Lw_\alpha \int _{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}u_h(x,y_j^\alpha )\partial _x\varphi _h(x,y_j^\alpha )dx +\lambda _y\sum \limits _{\alpha =1}^Lw_\alpha \int _{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}}v_h(x_i^\alpha ,y)\partial _y\varphi _h(x_i^\alpha ,y)dy\\&\quad =\lambda _x\sum \limits _{\alpha =1}^L\sum \limits _{\gamma =1}^Nw_\alpha \widehat{w}_\gamma u_{{\widehat{\gamma }},\alpha }\big (\sum \limits _{\beta =1}^N\varphi _{{\widehat{\beta }},\alpha }h_x^{(\beta )}(\widehat{x}_i^\gamma )\big )\varDelta x +\lambda _y\sum \limits _{\alpha =1}^L\sum \limits _{\gamma =1}^Nw_\alpha \widehat{w}_\gamma v_{\alpha ,{\widehat{\gamma }}}\big (\sum \limits _{\beta =1}^N\varphi _{\alpha ,{\widehat{\beta }}}h_y^{(\beta )}(\widehat{y}_j^\gamma )\big )\varDelta y. \end{aligned}\nonumber \\ \end{aligned}$$

(72)

Substituting Eq. (72) into the scheme Eq. (27), it gives

$$\begin{aligned} \begin{aligned}&{\overline{\varphi }}_{i,j}^{n+1}\\&\quad =\gamma _x\sum \limits _{\alpha =1}^Lw_\alpha \widehat{w}_1\big [\big (1-\frac{\lambda _x}{\gamma _x\widehat{w}_1}\sum \limits _{\gamma =1}^N\widehat{w}_\gamma u_{{\widehat{\gamma }},\alpha }h_x^{(1)} (\widehat{x}_i^\gamma )\varDelta x-\frac{\lambda _x}{\gamma _x\widehat{w}_1} u_{i-\frac{1}{2},\alpha }^+\big )\varphi _{i-\frac{1}{2},\alpha }^++\frac{\lambda _x}{\gamma _x\widehat{w}_1}\widehat{f}(\varphi _{i-\frac{1}{2},\alpha }^-,\varphi _{i-\frac{1}{2},\alpha }^+)\big ]\\&\qquad + \gamma _x\sum \limits _{\alpha =1}^L\sum \limits _{\beta =2}^{N-1}w_\alpha \widehat{w}_\beta \big (1-\frac{\lambda _x}{\gamma _x\widehat{w}_\beta }\sum \limits _{\gamma =1}^N \widehat{w}_\gamma u_{{\widehat{\gamma }},\alpha }h_x^{(\beta )}(\widehat{x}_i^\gamma )\varDelta x\big )\varphi _{{\widehat{\beta }},\alpha }\\&\qquad + \gamma _x\sum \limits _{\alpha =1}^Lw_\alpha \widehat{w}_N\big [\big (1-\frac{\lambda _x}{\gamma _x\widehat{w}_N}\sum \limits _{\gamma =1}^N\widehat{w}_\gamma u_{{\widehat{\gamma }},\alpha }h_x^{(N)} (\widehat{x}_i^\gamma )\varDelta x+\frac{\lambda _x}{\gamma _x\widehat{w}_N} u_{i+\frac{1}{2},\alpha }^-\big )\varphi _{i+\frac{1}{2},\alpha }^--\frac{\lambda _x}{\gamma _x\widehat{w}_N}\widehat{f}(\varphi _{i+\frac{1}{2},\alpha }^-,\varphi _{i+\frac{1}{2},\alpha }^+)\big ]\\&\qquad +\gamma _y\sum \limits _{\alpha =1}^Lw_\alpha \widehat{w}_1\big [\big (1-\frac{\lambda _y}{\gamma _y\widehat{w}_1}\sum \limits _{\gamma =1}^N\widehat{w}_\gamma v_{\alpha ,{\widehat{\gamma }}}h_y^{(1)} (\widehat{y}_j^\gamma )\varDelta y-\frac{\lambda _y}{\gamma _y\widehat{w}_1} v_{\alpha ,j-\frac{1}{2}}^+\big )\varphi _{\alpha ,j-\frac{1}{2}}^++\frac{\lambda _y}{\gamma _y\widehat{w}_1}\widehat{g}(\varphi _{\alpha ,j-\frac{1}{2}}^-,\varphi _{\alpha ,j-\frac{1}{2}}^+)\big ]\\&\qquad + \gamma _y\sum \limits _{\alpha =1}^L\sum \limits _{\beta =2}^{N-1}w_\alpha \widehat{w}_\beta \big (1-\frac{\lambda _y}{\gamma _y\widehat{w}_\beta }\sum \limits _{\gamma =1}^N \widehat{w}_\gamma v_{\alpha ,{\widehat{\gamma }}}h_y^{(\beta )}(\widehat{y}_j^\gamma )\varDelta y\big )\varphi _{\alpha ,{\widehat{\beta }}}\\&\qquad +\gamma _y\sum \limits _{\alpha =1}^Lw_\alpha \widehat{w}_N\big [\big (1-\frac{\lambda _y}{\gamma _y\widehat{w}_N}\sum \limits _{\gamma =1}^N\widehat{w}_\gamma v_{\alpha ,{\widehat{\gamma }}}h_y^{(N)} (\widehat{y}_j^\gamma )\varDelta y+\frac{\lambda _y}{\gamma _y\widehat{w}_N} v_{\alpha ,j+\frac{1}{2}}^-\big )\varphi _{\alpha ,j+\frac{1}{2}}^--\frac{\lambda _y}{\gamma _y\widehat{w}_N}\widehat{g}(\varphi _{\alpha ,j+\frac{1}{2}}^-,\varphi _{\alpha ,j+\frac{1}{2}}^+)\big ]. \end{aligned}\nonumber \\ \end{aligned}$$

(73)

Let us introduce the following formal formulations

$$\begin{aligned} \begin{aligned}&H_{x,\alpha }^{(1)}=\big (1-\frac{\lambda _x}{\gamma _x\widehat{w}_1}\sum \limits _{\gamma =1}^N\widehat{w}_\gamma u_{{\widehat{\gamma }},\alpha }h_x^{(1)} (\widehat{x}_i^\gamma )\varDelta x-\frac{\lambda _x}{\gamma _x\widehat{w}_1} u_{i-\frac{1}{2},\alpha }^+\big )\varphi _{i-\frac{1}{2},\alpha }^++\frac{\lambda _x}{\gamma _x\widehat{w}_1}\widehat{f}(\varphi _{i-\frac{1}{2},\alpha }^-,\varphi _{i-\frac{1}{2},\alpha }^+),\\&H_{x,\alpha }^{(\beta )}=\big (1-\frac{\lambda _x}{\gamma _x\widehat{w}_\beta }\sum \limits _{\gamma =1}^N \widehat{w}_\gamma u_{{\widehat{\gamma }},\alpha }h_x^{(\beta )}(\widehat{x}_i^\gamma )\varDelta x\big )\varphi _{{\widehat{\beta }},\alpha },\quad 2\le \beta \le N-1\\&H_{x,\alpha }^{(3)}=\big (1-\frac{\lambda _x}{\gamma _x\widehat{w}_N}\sum \limits _{\gamma =1}^N\widehat{w}_\gamma u_{{\widehat{\gamma }},\alpha }h_x^{(N)} (\widehat{x}_i^\gamma )\varDelta x+\frac{\lambda _x}{\gamma _x\widehat{w}_N} u_{i+\frac{1}{2},\alpha }^-\big )\varphi _{i+\frac{1}{2},\alpha }^--\frac{\lambda _x}{\gamma _x\widehat{w}_N}\widehat{f}(\varphi _{i+\frac{1}{2},\alpha }^-,\varphi _{i+\frac{1}{2},\alpha }^+). \end{aligned}\nonumber \\ \end{aligned}$$

(74)

Plugging the expression of $\widehat{f}(\cdot ,\cdot )$ Eq. (25) into Eq. (74), it gives

$$\begin{aligned} \begin{aligned}&H_{x,\alpha }^{(1)}=\big [1-\frac{\lambda _x}{\gamma _x\widehat{w}_1}\big (\sum \limits _{\gamma =1}^N\widehat{w}_\gamma u_{{\widehat{\gamma }},\alpha }h_x^{(1)} (\widehat{x}_i^\gamma )\varDelta x+\frac{1}{2} u_{i-\frac{1}{2},\alpha }^++\frac{1}{2}a_x\big )\big ]\varphi _{i-\frac{1}{2},\alpha }^++\frac{1}{2}\frac{\lambda _x}{\gamma _x\widehat{w}_1}(u_{i-\frac{1}{2},\alpha }^++a_x)\varphi _{i-\frac{1}{2},\alpha }^-,\\&H_{x,\alpha }^{(\beta )}=\big [1-\frac{\lambda _x}{\gamma _x\widehat{w}_\beta }\sum \limits _{\gamma =1}^N \widehat{w}_\gamma u_{{\widehat{\gamma }},\alpha }h_x^{(\beta )}(\widehat{x}_i^\gamma )\varDelta x\big ]\varphi _{{\widehat{\beta }},\alpha },\quad 2\le \beta \le N-1\\&H_{x,\alpha }^{(3)}=\big [1-\frac{\lambda _x}{\gamma _x\widehat{w}_N}\big (\sum \limits _{\gamma =1}^N\widehat{w}_\gamma u_{{\widehat{\gamma }},\alpha }h_x^{(N)} (\widehat{x}_i^\gamma )\varDelta x-\frac{1}{2} u_{i+\frac{1}{2},\alpha }^-+\frac{1}{2}a_x\big )\big ]\varphi _{i+\frac{1}{2},\alpha }^--\frac{1}{2}\frac{\lambda _x}{\gamma _x\widehat{w}_N}(u_{i+\frac{1}{2},\alpha }^--a_x)\varphi _{i+\frac{1}{2},\alpha }^+. \end{aligned}\nonumber \\ \end{aligned}$$

(75)

Then, it is easy to verify that under the CFL condition Eq. (42), $H_{x,\alpha }^{(1)}$, $H_{x,\alpha }^{(\beta )}$ $(2\le \beta \le N-1)$ and $H_{x,\alpha }^{(N)}$ are monotonically increasing with respect to their arguments, i.e.,

$$\begin{aligned} \frac{\partial H_{x,\alpha }^{(1)}(\varphi _{i-\frac{1}{2},\alpha }^-,\varphi _{i-\frac{1}{2},\alpha }^+)}{\partial \varphi _{i-\frac{1}{2},\alpha }^\pm }\ge 0, \quad \frac{\partial H_{x,\alpha }^{(\beta )}(\varphi _{{\widehat{\beta }},\alpha })}{\partial \varphi _{{\widehat{\beta }},\alpha }}\ge 0, \quad \frac{\partial H_{x,\alpha }^{(N)}(\varphi _{i+\frac{1}{2},\alpha }^-,\varphi _{i+\frac{1}{2},\alpha }^+)}{\partial \varphi _{i+\frac{1}{2},\alpha }^\pm }\ge 0.\nonumber \\ \end{aligned}$$

(76)

Moreover, we have

$$\begin{aligned} \begin{aligned}&\widehat{w}_1H_{x,\alpha }^{(1)}(m,m)+\sum \limits _{\beta =2}^{N-1}\widehat{w}_\beta H_{x,\alpha }^{(\beta )}(m)+\widehat{w}_3H_{x,\alpha }^{(N)}(m,m)=m,\\&\widehat{w}_1H_{x,\alpha }^{(1)}(M,M)+\sum \limits _{\beta =2}^{N-1}\widehat{w}_\beta H_{x,\alpha }^{(\beta )}(M)+\widehat{w}_3H_{x,\alpha }^{(N)}(M,M)=M. \end{aligned}\nonumber \\ \end{aligned}$$

(77)

Similar results can be obtained for

$$\begin{aligned} \begin{aligned}&H_{\alpha ,y}^{(1)}=\big [1-\frac{\lambda _y}{\gamma _y\widehat{w}_1}\sum \limits _{\gamma =1}^N\widehat{w}_\gamma v_{\alpha ,{\widehat{\gamma }}}h_y^{(1)} (\widehat{y}_j^\gamma )\varDelta y-\frac{\lambda _y}{\gamma _y\widehat{w}_1} v_{\alpha ,j-\frac{1}{2}}^+\big ]\varphi _{\alpha ,j-\frac{1}{2}}^++\frac{\lambda _y}{\gamma _y\widehat{w}_1}\widehat{g}(\varphi _{\alpha ,j-\frac{1}{2}}^-,\varphi _{\alpha ,j-\frac{1}{2}}^+),\\&H_{\alpha ,y}^{(2)}=\big [1-\frac{\lambda _y}{\gamma _y\widehat{w}_\beta }\sum \limits _{\gamma =1}^N \widehat{w}_\gamma v_{\alpha ,{\widehat{\gamma }}}h_y^{(\beta )}(\widehat{y}_j^\gamma )\varDelta y\big ]\varphi _{\alpha ,{\widehat{\beta }}},\\&H_{\alpha ,y}^{(3)}=\big [1-\frac{\lambda _y}{\gamma _y\widehat{w}_N}\sum \limits _{\gamma =1}^N\widehat{w}_\gamma v_{\alpha ,{\widehat{\gamma }}}h_y^{(N)} (\widehat{y}_j^\gamma )\varDelta y+\frac{\lambda _y}{\gamma _y\widehat{w}_N} v_{\alpha ,j+\frac{1}{2}}^-\big ]\varphi _{\alpha ,j+\frac{1}{2}}^--\frac{\lambda _y}{\gamma _y\widehat{w}_N}\widehat{g}(\varphi _{\alpha ,j+\frac{1}{2}}^-,\varphi _{\alpha ,j+\frac{1}{2}}^+). \end{aligned}\nonumber \\ \end{aligned}$$

(78)

Therefore, ${\overline{\varphi }}_{i,j}^{n+1}\in [m,M]$ under the CFL condition Eq. (42) since it is a convex combination of all the points values involved.