Download PDF by Dmitri Kuzmin: A Guide to Numerical Methods for Transport Equations

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Extra info for A Guide to Numerical Methods for Transport Equations

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In particular, this kind of approximation provides a natural treatment of inviscid fluxes in conservation laws. In many situations, it turns out to be more accurate than the independent approximation of the involved variables [109, 110]. The group finite element approximation of the convective flux is given by (vu)h = ∑(v j u j )ϕ j . 10) j Obviously, this function is easier to differentiate than the product vh uh . 5) is approximated by ∇ · (vu)h = ∑ u j (v j · ∇ϕ j ). 11) j That is, the conservative semi-discrete form of the convective term is the sum of nodal values multiplied by the convective derivatives of the basis functions.

21), the i−th row sum of the consistent mass matrix MC = {mi j } is ∑ mi j = j Ω ϕi ∑ ϕ j dx = j Ω ϕi dx. 1 Discretization on Unstructured Meshes 47 Furthermore, summation over all indices yields the volume (area) of the domain Ω ∑ ∑ mi j = i ∑ ϕi dx = |Ω |. 24) ∑ ∇ϕi · ∇ϕ j dx = 0. 26) j ∑ ϕi ∇ϕ j dx = Ω i Ω ∇ϕ j dx. 3. It ensures that the result of numerical differentiation equals zero if all nodal values of the numerical solution coincide and define a constant field. By definition, finite element basis functions have compact support.

Thus, even solutions to rather simple problems may behave in an unpredictable manner. A finite difference scheme proves conservative if it can be written in terms of numerical fluxes from one grid point into another. Finite volume and discontinuous Galerkin methods are conservative by construction, both globally and locally. The continuous Galerkin FEM provides global conservation [134] and is claimed to be locally conservative by some authors [158, 162]. Conservation may be lost if inaccurate quadrature rules or nonstandard approximations are employed.

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A Guide to Numerical Methods for Transport Equations by Dmitri Kuzmin

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