Upwind Schemes for Scalar Advection-Dominated Problems in the Discrete Exterior Calculus

Abstract

We present the discrete exterior calculus (DEC) to solve discrete partial differential equations on discrete objects such as cell complexes. To cope with advection-dominated problems, we introduce a novel stabilization technique to the DEC. To this end, we use the fact that the DEC coincides in special situations with known discretization schemes such as finite volumes or finite differences. Thus, we can carry over well-established upwind stabilization methods introduced for these classical schemes to the DEC. This leads in particular to a stable discretization of the Lie-derivative. We present the numerical features of this new discretization technique and study its numerical properties for simple model problems and for advection-diffusion processes on simple surfaces.

Appendix: Derivation of the Discrete Contraction Formula

In this section, we give a formal derivation of the formula for the discrete contraction. In contrast to [15, p. 93] we obtain slightly different normalization factors. For the definitions see [15].

Theorem 3

For piecewise constant differential forms, ω ^k+1 is the evaluation of $\mathit{\text{i}}_{\boldsymbol{\sigma }^{1}}$ on a simplex σ ^k with the algebraic definition from the geometric definition using the extrusion of σ ^k equivalent to the evaluation of the discrete differential form on σ ^k+1.

Proof

We assume the differential form ω ^k+1 as piecewise constant on the k + 1-simplices which yields

$$\displaystyle{\left \langle \omega ^{k+1},\text{extr}_{\boldsymbol{\sigma }^{ 1}}(\sigma ^{k},t)\right \rangle = \frac{\left \vert \text{extr}_{\boldsymbol{\sigma }^{1}}(\sigma ^{k},t)\right \vert } {\left \vert \sigma ^{k+1}\right \vert } \left \langle \omega ^{k+1},\sigma ^{k+1}\right \rangle.}$$

Hence, we obtain

$$\displaystyle\begin{array}{rcl} \left \langle \text{i}_{v}\omega ^{k+1},\sigma ^{k}\right \rangle & &:= \left. \frac{\text{d}} {\text{dt}}\right \vert _{t=0}\left \langle \omega ^{k+1},\text{extr}(\sigma ^{k},v_{ t})\right \rangle {}\\ & & = \left. \frac{\text{d}} {\text{dt}}\right \vert _{t=0}\left (\left \vert \text{extr}_{\boldsymbol{\sigma }^{1}}(\sigma ^{k},t)\right \vert \right ) \frac{1} {\left \vert \sigma ^{k+1}\right \vert }\left \langle \omega ^{k+1},\sigma ^{k+1}\right \rangle. {}\\ \end{array}$$

This shows that we have to consider the map $t\mapsto \left \vert \text{extr}_{\boldsymbol{\sigma }^{1}}(\sigma ^{k},t)\right \vert$.

Following [15, Chap. 8.3], we get

$$\displaystyle{\left \vert \text{extr}_{\boldsymbol{\sigma }^{1}}(\sigma ^{k},t)\right \vert = \left \vert \sigma ^{k+1}\right \vert \frac{\left \vert h(t)\right \vert } {\left \vert h(1)\right \vert },}$$

where σ ^k is the base side of a (k + 1)-simplex and h its height. Here, we use the geometric definition and define the extrusion $\text{extr}_{\boldsymbol{\omega }^{1 }}(\sigma ^{k},t)$ of a face ω ^k as the (k + 1)-simplex spanned by the face and a vector $\dot{\mathbf{x}}(t) =\boldsymbol{\sigma } ^{1}(1 - t),0 \leq t \leq 1$. We assume without loss of generality that the common vertex σ ⁰ of the face and the edge lies at the origin. Because we have $\dot{\mathbf{x}}(0) = 0$, we get $\mathbf{x}(t) = t -\frac{t^{2}} {2}$. The height is proportional to the edge $\boldsymbol{\sigma }^{1} = \mathbf{x}(1)$, spanning the (k + 1)-simplex together with the k-simplex σ ^k. Thus, we have for 0 < t < 1

$$\displaystyle\begin{array}{rcl} \frac{h(t)} {h(1)} = \frac{\vert \mathbf{x}(t)\vert } {\vert \mathbf{x}(1)\vert } = \frac{\vert \boldsymbol{\sigma }^{1}(t -\frac{t^{2}} {2} )\vert } {\vert \boldsymbol{\sigma }^{1}\vert } = t -\frac{t^{2}} {2}.& &{}\end{array}$$

(6.34)

Hence, we obtain

$$\displaystyle{\left. \frac{\text{d}} {\text{dt}}\right \vert _{t=0}\left (\left \vert \text{extr}_{\boldsymbol{\sigma }^{1}}(\sigma ^{k},t)\right \vert \right ) = 1\left \vert \sigma ^{k+1}\right \vert.}$$

This yields the claim. □