Geometric error of finite volume schemes for conservation laws on evolving surfaces

Abstract

This paper studies finite volume schemes for scalar hyperbolic conservation laws on evolving hypersurfaces of $\mathbb {R}^3$. We compare theoretical schemes assuming knowledge of all geometric quantities to (practical) schemes defined on moving polyhedra approximating the surface. For the former schemes error estimates have already been proven, but the implementation of such schemes is not feasible for complex geometries. The latter schemes, in contrast, only require (easily) computable geometric quantities and are thus more useful for actual computations. We prove that the difference between approximate solutions defined by the respective families of schemes is of the order of the mesh width. In particular, the practical scheme converges to the entropy solution with the same rate as the theoretical one. Numerical experiments show that the proven order of convergence is optimal.

Appendix

Here we give the proof of Lemma 8.

Proof

It is sufficient to show

$$\begin{aligned} \Big | \frac{d}{dt} \frac{|K(t)|}{ |\bar{K}(t)|} \Big | \le Ch. \end{aligned}$$

(63)

We have

$$\begin{aligned} \frac{d}{dt} \frac{|K(t)|}{|\bar{K}(t)|}&= \frac{d}{dt} \frac{|K(t)|- |\bar{K}(t)| }{|\bar{K}(t)|} \nonumber \\&= \frac{(|K(t)|- |\bar{K}(t)| )_t }{|\bar{K}(t)|} + \frac{1}{ |\bar{K}(t)|} \Big ( \frac{|K(t)|}{ |\bar{K}(t)|} -1 \Big ) |\bar{K}(t)|_t \end{aligned}$$

(64)

so that, due to Lemma 3, it suffices to show

$$\begin{aligned} \Big |\frac{(|K(t)|- |\bar{K}(t)| )_t }{ |\bar{K}(t)|} \Big |\le C h \quad \text { and } \quad \big | |\bar{K}(t)|_t\big | \le Ch. \end{aligned}$$

(65)

The estimate (65)$_2$ is immediate, so we turn our attention to (65)$_1$. Let us assume $t \in [t_n,t_{n+1}]$ and let $\bar{K}(t_n)$ be the convex hull of the vertices $v_0:=(0,0,0),\, v_1:= (h,0,0),\,v_2:= (x,y,0)$. We define

$$\begin{aligned} \varPhi ^n(\cdot , t) : \varGamma (t_n) \rightarrow \varGamma (t), \quad \varPhi ^n(\cdot ,t):= \varPhi (\cdot ,t) \circ \varPhi (\cdot ,t_n)^{-1}, \end{aligned}$$

such that $\varPhi ^n(\cdot ,t_n)$ is the identity map. We denote the canonical projection $\bar{K}(t_n) \rightarrow K(t_n)$ by $c$ and abbreviate $\varPhi ^n \circ c$ by $\varPhi ^n_c$. The scaled directional derivatives are denoted $\partial _{v_1} := h \partial _{x_1}$ and $\partial _{v_2}:= x\partial _{x_1} + y\partial _{x_2}.$ Then,

$$\begin{aligned} (|K(t)|- |\bar{K}(t)| )_t&= \frac{d}{dt} \left( \frac{1}{hy} \int \limits _{\bar{K}(t_n)} \Vert \partial _{v_1} \varPhi ^n_c(x,t) \times \partial _{v_2} \varPhi ^n_c(x,t)\Vert \, d\bar{K}(t_n) \right. \nonumber \\&\left. -\frac{1}{2} \Vert (\varPhi ^n(v_1,t) - \varPhi ^n(v_0,t)) \times (\varPhi ^n(v_2,t) - \varPhi ^n(v_0,t))\Vert \right) \nonumber \\ \end{aligned}$$

(66)

and thus

$$\begin{aligned}&\big | (|K(t)|- |\bar{K}(t)| )_t\big | \nonumber \\&\le \frac{1}{hy} \left( \int \limits _{\bar{K}(t_n)} \big \Vert \partial _t \partial _{v_1} \varPhi ^n_c(x,t) \times \partial _{v_2} \varPhi ^n_c(x,t) + \partial _{v_1} \varPhi ^n_c(x,t) \times \partial _t\partial _{v_2} \varPhi ^n_c(x,t) \right. \nonumber \\&\quad \left. - \partial _t(\varPhi ^n(v_1,t) - \varPhi ^n(v_0,t)) \times (\varPhi ^n(v_2,t) - \varPhi ^n(v_0,t)) \right. \nonumber \\&\quad \left. - (\varPhi ^n(v_1,t) - \varPhi ^n(v_0,t)) \times \partial _t(\varPhi ^n(v_2,t)\right. \nonumber \\&\quad \left. - \varPhi ^n(v_0,t)) \big \Vert \, d\bar{K}(t_n)\right) . \end{aligned}$$

(67)

Using the mean value theorem this implies

$$\begin{aligned}&\Big |2 (|K(t)|- |\bar{K}(t)| )_t\Big | \nonumber \\&\quad \le \big \Vert \partial _t \partial _{v_1} \varPhi ^n_c(\xi _1,t) \times \partial _{v_2} \varPhi ^n_c(\xi _1,t) + \partial _{v_1} \varPhi ^n_c(\xi _2,t) \times \partial _t\partial _{v_2} \varPhi ^n_c(\xi _2,t) \nonumber \\&\quad \quad - \partial _{v_1} \partial _t\varPhi ^n(\xi _3,t) \times \partial _{v_2} \varPhi ^n(\xi _4,t) - \partial _{v_1} \varPhi ^n(\xi _5,t) \times \partial _{v_2} \partial _t\varPhi ^n(\xi _6,t) \big \Vert \quad \quad \end{aligned}$$

(68)

for some $\xi _1,\dots \xi _6 \in \bar{K}(t_n)$. We have

$$\begin{aligned} ( \partial _t \partial _{x_1} \varPhi _c^n)(\xi _1,t) = D( \partial _t \varPhi ^n)( c(\xi _1),t) \partial _{x_1} c(\xi _1) = (\partial _{x_1} \partial _t \varPhi ^n)( \xi _3,t) + \mathcal {O}(h)\quad \quad \end{aligned}$$

(69)

because of (23) and the regularity of $\varPhi ^n$. This, and a straightforward estimate for the second factor in the vector product, leads to

$$\begin{aligned} \partial _t \partial _{v_1} \varPhi ^n_c(\xi _1,t) \times \partial _{v_2} \varPhi ^n_c(\xi _1,t) - \partial _{v_1}\partial _t\varPhi ^n(\xi _3,t) \times \partial _{v_2} \varPhi ^n(\xi _4,t) = \mathcal {O}(h^3). \end{aligned}$$

(70)

Using a similar estimate for the remaining terms in (68) we find

$$\begin{aligned} (|K(t)|- |\bar{K}(t)| )_t = \mathcal {O}(h^3) \end{aligned}$$

(71)

which implies (65)$_1$ because of (5).$\square$