Second-order accurate finite volume method for G-equation on polyhedral meshes

Abstract

In this paper, we propose a cell-centered finite volume method to numerically solve the G-equation on polyhedral meshes in three-dimensional space, that is, a general type of the level-set equation including advective, normal, and mean curvature flow motions. The main contribution is to design a numerical algorithm for the regularized mean curvature flow equation that can be consistently combined regarding the size of the time step with previous algorithms for the advective and normal flows on polyhedral meshes. For a spatial discretization, we use a flux-balanced approximation with an orthogonal splitting of displacement vector from a center of the cell to a center of the face. For a temporal discretization, we use a nonlinear Crank–Nicolson method with a deferred correction method which gives us, firstly, second-order accuracy in space and time similarly to the algorithms for the advective and normal flow equations, and, secondly, a possibility of straightforward domain decomposition for efficient parallel computation. Numerical experiments quantitatively show that the size of time step proportional to an average size of computational cells is enough to obtain the second-order convergence in space and time for smooth solutions of the general level set equation. A qualitative comparison is presented for a nontrivial example to compare numerical results obtained with hexahedral and polyhedral meshes. Finally, an example of solving numerically the G-equation used in combustion literature is given.

Appendix A: Iterative IIOE method

In order to complete the finite volume method to solve (2), we briefly review the iterative inflow-implicit outflow-explicit method (IIOE) [24, 25] of discretizing the advective and normal flow equations which appear in the second and third term of (2):

$$\begin{aligned} \frac{\partial \phi }{\partial t} + {\textbf{u}} \cdot \nabla \phi = 0 . \end{aligned}$$

(A1)

The following two forms of the velocity function ${\textbf{u}}$ are considered:

$$\begin{aligned} {\textbf{u}} = {\textbf{v}}({\textbf{x}}, t) \quad \text {or} \quad {\textbf{u}} = \delta \frac{\nabla \phi }{|\nabla \phi |}, \quad ({\textbf{x}}, t) \in \Omega \times [0,T] , \end{aligned}$$

(A2)

that describes the advective and normal flow, respectively, for an evolution of surface described by a zero level set of $\phi$. Without a loss of generality, we choose the spatial variable $\delta = \delta ({\textbf{x}})$ as the constant $\delta = 1$ in (4) for the rest of derivation of the method.

Firstly, we introduce a spatial discretization, and, secondly, we discuss a temporal discretization. Let us denote the set of indices $\bar{{\mathcal {F}}}_p$ and $\bar{{\mathcal {B}}}_p$ of triangles obtained by a tessellation of internal and boundary faces $e_f$, $f \in {\mathcal {F}}_p$ and $e_b$, $b \in {\mathcal {B}}_p$, respectively; see more details how to construct the tessellation in [25]. For a triangle index $i \in \bar{{\mathcal {F}}}_p$, $\bar{e}_i$ is the triangle as a subset of some face $e_f$, $f \in {\mathcal {F}}_p$. Furthermore, $\bar{{\textbf{n}}}_{pi}$ is the outward normal vector to the triangle whose length is the area of $\bar{e}_i$, and $\bar{{\textbf{x}}}_i$ is the center of the triangle. Using the Gauss divergence theorem on (A1), the spatial discretization is obtained:

$$\begin{aligned} \int _{\Omega _p} \frac{\partial \phi }{\partial t} + \sum _{i \in \bar{{\mathcal {F}}}_{p} \cup \bar{{\mathcal {B}}}_{p} } a_{pi} \left( \phi _{pi} - \phi _p \right) = 0, \quad a_{pi} \equiv \int _{\bar{e}_{i}} {\textbf{u}} \cdot \bar{{\textbf{n}}}_{pi} \approx {\textbf{u}}\left( \bar{{\textbf{x}}}_i, t \right) \cdot \bar{{\textbf{n}}}_{pi}, \end{aligned}$$

(A3)

where $\phi _{pi}$ is an approximated value defined at the center of the triangle $\bar{{\textbf{x}}}_i$. Note that the flux $a_{pi}$ is computed using Gaussian quadrature of degree 1 which insures the exactness of integration as polynomials of degree 1.

A crucial point in (A3) is how to approximate the value $\phi _{pi}$. In order to obtain a second-order accurate upwind scheme, we prepare some values at vertices and gradients at centers of tessellated triangles in the following procedure. A value $\phi ({\textbf{x}}_v)$ at an internal vertex is computed by an inverse distance weighted average from adjacent cells. For the fixed vertex ${\textbf{x}}_v$, let us denote a set of indices of cells containing ${\textbf{x}}_v$ as a vertex, ${\mathcal {N}}_v \equiv \left\{ p \in {\mathcal {I}}: {\textbf{x}}_v \in \partial \Omega _p \right\} .$ Then the value at internal vertex is obtained by the inverse distance weighted average of the first order Taylor polynomial:

$$\begin{aligned} \phi ({\textbf{x}}_v) = \frac{ \sum _{p \in {\mathcal {N}}_v } |{\textbf{d}}_{pv} |^{-1} (\phi ({\textbf{x}}_p) + \nabla \phi ({\textbf{x}}_p) \cdot {\textbf{d}}_{pv} ) }{ \sum _{p \in {\mathcal {N}}_v } |{\textbf{d}}_{pv} |^{-1} }. \end{aligned}$$

Note that a value at boundary vertex is directly assigned by Dirichlet boundary condition. From the obtained vertex values, a gradient at a center of tessellated triangle is computed. For an internal triangle $\bar{e}_i \subset e_f$, $f\in {\mathcal {F}}_p$, there exists $q \in {\mathcal {I}}$ such that $e_f \subset \partial \Omega _p \cap \partial \Omega _q$. Then, we consider two tetrahedrons whose apices are ${\textbf{x}}_p$ and ${\textbf{x}}_q$ and they have the common base $\bar{e}_i$. We denote the set of all vertices of the tetrahedrons as ${\mathcal {P}}_i$. Then, the gradient $\bar{\varvec{\beta }}_i$ at the center of $\bar{e}_i$ is computed by the weighted least-squares minimization:

$$\begin{aligned} (\bar{\alpha }_i, \bar{\varvec{\beta }}_i) = {\mathop {\mathrm{arg\,min}}\limits _{(a_i, {\textbf{b}}_i) \in {\mathbb {R}}^4}} \sum _{{\textbf{y}} \in {\mathcal {P}}_i} w_i({\textbf{y}}) \big |a_i + {\textbf{b}}_i \cdot ({\textbf{x}} - \bar{{\textbf{x}}}_i) - \phi ({\textbf{x}}) \big |^2, \end{aligned}$$

(A4)

where $\bar{{\textbf{x}}}_i$ is the center of $\bar{e}_i$ the weight function is defined by $w_i({\textbf{y}}) = |{\textbf{y}} - \bar{{\textbf{x}}}_i|^{-2}$. The formula (A4) is a generalization of the diamond-cell method described for a regular structured hexahedron cell in [20]. Now, we define a so-called average-based gradient [19] as the inverse distance weighted average of gradients:

$$\begin{aligned} {\mathcal {D}}_p \phi = \frac{ \sum _{i \in \bar{{\mathcal {F}}}_p } |{\textbf{d}}_{pi}|^{-1} \bar{\varvec{\beta }}_i }{ \sum _{i \in \bar{{\mathcal {F}}}_p } |{\textbf{d}}_{pi}|^{-1}}, \quad {\textbf{d}}_{pi} = {\textbf{x}}_p - \bar{{\textbf{x}}}_i. \end{aligned}$$

(A5)

Finally, we compute the value $\phi _{pi}$ at an internal triangle $\bar{e}_i \subset e_f$, $f\in {\mathcal {F}}_p$, in (A3) using the average-based gradient and the upwind principle:

$$\begin{aligned} \begin{aligned} \phi _{pi} = {\left\{ \begin{array}{ll} \phi _p + {\mathcal {D}}_p \phi \cdot (\bar{{\textbf{x}}}_i - {\textbf{x}}_p) &{}\quad \text {if } \,\, a_{pi} \ge 0,\\ \phi _q + {\mathcal {D}}_q \phi \cdot (\bar{{\textbf{x}}}_i - {\textbf{x}}_q) &{}\quad \text {if } \,\, a_{pi} < 0 , \end{array}\right. } \end{aligned} \end{aligned}$$

(A6)

where the neighbor cell index $q \in {\mathcal {I}}$ is such that $e_f \subset \partial \Omega _p \cap \partial \Omega _q$. The value $\phi _{pi}$ at a boundary triangle $\bar{e}_i \subset e_b$, $b\in {\mathcal {B}}_p$, in (A3) is computed by

$$\begin{aligned} \begin{aligned} \phi _{pi} = {\left\{ \begin{array}{ll} \phi _p + {\mathcal {D}}_p \phi \cdot (\bar{{\textbf{x}}}_i - {\textbf{x}}_p) &{}\quad \text {if } \,\, a_{pi} \ge 0,\\ \phi _{bi} &{}\quad \text {if } \,\, a_{pi} < 0, \end{array}\right. } \end{aligned} \end{aligned}$$

(A7)

where $\phi _{bi} \equiv \phi _b(\bar{{\textbf{x}}}_i)$ from Dirichlet boundary condition. We finally obtain the spatial discretization from (A6) and (A7):

$$\begin{aligned} \begin{aligned} \int _{\Omega _p} \frac{\partial \phi }{\partial t} =&-\sum _{i \in \bar{{\mathcal {F}}}^{-}_{p}} \left( \phi _q + {\mathcal {D}}_q \phi \cdot {\textbf{d}}_{qi} - \phi _p \right) a_{pi} - \sum _{i \in \bar{{\mathcal {F}}}^{+}_{p}} \left( {\mathcal {D}}_p \phi \cdot {\textbf{d}}_{pi} \right) a_{pi} \\&- \sum _{i \in \bar{{\mathcal {B}}}^{-}_{p}} \left( \phi _{bi} - \phi _p \right) a_{pi} - \sum _{i \in \bar{{\mathcal {B}}}^{+}_{p}} \left( {\mathcal {D}}_p \phi \cdot {\textbf{d}}_{pi} \right) a_{pi}, \end{aligned} \end{aligned}$$

(A8)

where $\bar{{\mathcal {F}}}^{-}_{p}$ and $\bar{{\mathcal {B}}}^{-}_{p}$ are the subsets of $\bar{{\mathcal {F}}}_p$ and $\bar{{\mathcal {B}}}_p$ with $a_{pi} < 0$, respectively, and $\bar{{\mathcal {F}}}^{+}_{p} \equiv \bar{{\mathcal {F}}}_p {\setminus } \bar{{\mathcal {F}}}^{-}_{p}$, $\bar{{\mathcal {B}}}^{+}_{p} \equiv \bar{{\mathcal {B}}}_p {\setminus } \bar{{\mathcal {B}}}^{-}_{p}$.

Concerning the time discretization, we apply the deferred correction method because of the same reasons as discussed in Sect. 3. Following the analogous notations as in previous sections, the IIOE method inspired by [20,21,22,23] is defined by using an implicit and explicit time discretization of terms in (A8) on an inflow and outflow triangle, respectively:

$$\begin{aligned} \begin{aligned} \frac{|\Omega _p|}{\Delta t} \left( \phi _p^{n,k} - \phi _p^{n-1} \right) =&- \sum _{i \in \bar{{\mathcal {F}}}^{-}_{p}} \left( \phi _q^{n,k} + {\mathcal {D}}_q \phi ^{n,k-1} \cdot {\textbf{d}}_{qi} - \phi _p^{n,k} \right) a_{pi}^{n-1} \\&- \sum _{i \in \bar{{\mathcal {B}}}^{-}_{p}} \left( \phi _{bi}^{n} - \phi _p^{n,k} \right) a_{pi}^{n-1} - \sum _{i \in \bar{{\mathcal {B}}}^{+}_{p} \cup \bar{{\mathcal {F}}}^{+}_{p}} \left( {\mathcal {D}}_p \phi ^{n-1} \cdot {\textbf{d}}_{pi} \right) a_{pi}^{n-1}, \end{aligned} \end{aligned}$$

(A9)

where $\phi ^{n,0} = \phi ^{n-1}$. Note that the average-based gradient ${\mathcal {D}}_p \phi ^{n,k-1}$ is computed by values at centers of cells from $\phi ^{n,k-1}$ and values at centers of boundary faces from the nth time level.