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

Michael Griebel; Christian Rieger; Alexander Schier

doi:10.1007/978-3-319-56602-3_6

Transport Processes at Fluidic Interfaces pp 145-175 | Cite as

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

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Michael Griebel
Christian Rieger
Alexander Schier

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First Online: 14 July 2017

Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)

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.

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Acknowledgements

M. Griebel and A. Schier thank the DFG for the financial support through the Priority Programme 1506: Transport Processes at Fluidic Interfaces (SPP 1506). The authors would also like to thank B. Zwicknagl for reading parts of the manuscript and for inspiring discussions.

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+1is the evaluation of$\mathit{\text{i}}_{\boldsymbol{\sigma }^{1}}$on a simplex σ^kwith the algebraic definition from the geometric definition using the extrusion of σ^kequivalent to the evaluation of the discrete differential form on σ^k+1.

Proof

We assume the differential formω^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 σ⁰ 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. □

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Michael Griebel
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Christian Rieger
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Alexander Schier
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1.Institute for Numerical SimulationUniversity of BonnBonnGermany

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

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Notes

Acknowledgements

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