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

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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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... Discrete Exterior Calculus (DEC) is a relatively new numerical method for solving partial differential equations (PDE's), based on the discretization of the Exterior Differential Calculus theory of Cartan [2]. It was first proposed by A. Hirani in his PHD thesis [7] and has been applied to solve Darcy's equation [6], Navier-Stokes and Poisson's equations [10], plasticity and failure of isotropic materials [4], and the transport equation with incompressible flow for advection-dominated problems [5]. In particular, the authors of [5] showed that, in simple cases, the system of equations resulting from DEC are equivalent to other numerical methods, such as the Finite Differences and Finite Volume methods, leading to a stable upwind DEC variation, which involves the Lie derivative. ...
... It was first proposed by A. Hirani in his PHD thesis [7] and has been applied to solve Darcy's equation [6], Navier-Stokes and Poisson's equations [10], plasticity and failure of isotropic materials [4], and the transport equation with incompressible flow for advection-dominated problems [5]. In particular, the authors of [5] showed that, in simple cases, the system of equations resulting from DEC are equivalent to other numerical methods, such as the Finite Differences and Finite Volume methods, leading to a stable upwind DEC variation, which involves the Lie derivative. ...
... In this paper, we propose a local DEC discretization of the Convection-Diffusion equation for compressible and incompressible flow. This formulation is developed based on DEC and "natural" geometric arguments, assuming the values of the particle velocity field are given at the nodes of the primal mesh, thus making it different form the discretization in [5]. Since our discretization on the Convection-Diffusion equation will not involve discrete dual-to-primal Hodge star operators, it can be developed in a local manner similar to that of FEML (see [12]) together with its assembling technique, leading to an efficient implementation. ...
Chapter
A discretization of the Convection-Diffusion equation is developed based on Discrete Exterior Calculus (DEC). While DEC discretization of the diffusive term in the equation is well understood, the convective part (with non-constant convective flow) had not been DEC discretized. In this study, we develop such discretization of the convective term using geometric arguments. We can discretize the convective term for both compressible and incompressible flow. Moreover, since the Finite Element Method with linear interpolation functions (FEML) and DEC local matrix formulations are similar, this numerical scheme is well suited for parallel computing. Using this feature, numerical tests are carried out on simple domains with coarse and fine meshes to compare DEC and FEML and show numerical convergence for stationary problems.
... The aim of DEC is to solve partial differential equations preserving their geometrical and physical features as much as possible. There are only a few papers about implementions of DEC to solve certain PDEs, such as the Darcy flow and Poisson's equation [8], the Navier-Stokes equations [9], the simulation of elasticity, plasticity and failure of isotropic materials [4], some comparisons with the finite differences and finite volume methods on regular flat meshes [6], as well as applications in digital geometry processing [3]. ...
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We present a local formulation for 2D Discrete Exterior Calculus (DEC) similar to that of the Finite Element Method (FEM), which allows a natural treatment of material heterogeneity (element by element). It also allows us to deduce, in a robust manner, anisotropic fluxes and the DEC discretization of the pullback of 1-forms by the anisotropy tensor, i.e. we deduce how the anisotropy tensor acts on primal 1-forms. Due to the local formulation, the computational cost of DEC is similar to that of the Finite Element Method with Linear interpolations functions (FEML). The numerical DEC solutions to the anisotropic Poisson equation show numerical convergence, are very close to those of FEML on fine meshes and are slightly better than those of FEML on coarse meshes.
... Perhaps the first numerical application of DEC to PDE was given in [9] in order to solve Darcy flow and Poisson's equation. In [7], the authors develop a modification of DEC and show that in simple cases (e.g. flat geometry and regular meshes), the equations resulting from DEC are equivalent to classical numerical schemes such as finite difference or finite volume discretizations. ...
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We revisit the theory of Discrete Exterior Calculus (DEC) in 2D relying only on Vector Calculus and Matrix Algebra, and develop a formulation of DEC that applies to general triangulations, thus removing the restriction of well-centered meshes. This formulation is tested by solving the Poisson equation numerically and comparing the solutions against those found using the Finite Element Method with linear elements (FEML). Numerical convergence of DEC is also illustrated.
... Perhaps the first numerical application of DEC to PDE was given in [8] in order to solve Darcy flow and Poisson's equation. In [6], the authors develop a modification of DEC and show that in simple cases (e.g. flat geometry and regular meshes), the equations resulting from DEC are equivalent to classical numerical schemes such as finite difference or finite volume discretizations. ...
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We revisit the theory of Discrete Exterior Calculus (DEC) in 2D, a method for solving partial differential equations. We develop a local formulation of DEC that applies to general triangulations, thus removing the restriction of well-centered meshes. The method is tested on the Poisson equation, and we show that for linear triangular elements, the global matrix of the linear system is the same for FEM and DEC, whereas the vector of independent terms is different. This work is also an introduction to DEC relying only on Vector Calculus and Matrix Algebra.
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A conservative discretization of incompressible Navier-Stokes equations on simplicial meshes is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the contraction operator and a combinatorial discretization of the wedge product. The governing equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. The discretization is then carried out by substituting with the corresponding discrete operators based on the DEC framework. Numerical experiments reveal a second order accuracy for the developed scheme when using structured-triangular meshes, and first order accuracy for otherwise unstructured meshes. By construction, the method is conservative in that both mass and vorticity are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step.
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We derive a numerical method for Darcy flow, hence also for Poisson's equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedral meshes. DEC is a coordinate invariant discretization, in that it does not depend on the embedding of the simplices or the whole mesh. We start by rewriting the governing equations of Darcy flow using the language of exterior calculus. This yields a formulation in terms of flux differential form and pressure. The numerical method is then derived by using the framework provided by DEC for discretizing differential forms and operators that act on forms. We also develop a discretization for spatially dependent Hodge star that varies with the permeability of the medium. This also allows us to address discontinuous permeability. The matrix representation for our discrete non-homogeneous Hodge star is diagonal, with positive diagonal entries. The resulting linear system of equations for flux and pressure are saddle type, with a diagonal matrix as the top left block. The performance of the proposed numerical method is illustrated on many standard test problems. These include patch tests in two and three dimensions, comparison with analytically known solution in two dimensions, layered medium with alternating permeability values, and a test with a change in permeability along the flow direction. We also show numerical evidence of convergence of the flux and the pressure. A convergence experiment is also included for Darcy flow on a surface. A short introduction to the relevant parts of smooth and discrete exterior calculus is included in this paper. We also include a discussion of the boundary condition in terms of exterior calculus.
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