By Dmitri Kuzmin
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Extra resources for A Guide to Numerical Methods for Transport Equations
K. The value of the approximate solution at node i and time level t n is denoted by uni ≈ ui (t n ). 14), the value of u0i = ui (0) is assumed to be known. In principle, the time can be treated just like an extra space dimension. Finite difference, finite volume, and finite element methods are readily applicable to functions of x = (x1 , . . , xd ,t) ∈ Rd+1 defined in the space-time domain Ω × (0, T ). However, simultaneous computation of uni for all nodes and time levels is often too expensive.
1 Discretization on Unstructured Meshes 45 In the Galerkin method, the weighting functions wh are taken from the same finitedimensional space as the basis functions. 12) with wh = ϕi yields the following equation for nodal value ui ∑ j Ω ϕi ϕ j dx du j +∑ dt j = Ω Ω (ϕi (v j · ∇ϕ j ) + ∇ϕi · (ε ∇ϕ j )) dx u j ϕi s dx − ΓN ϕi h ds. 14) The value of the flux h = −(ε ∇u) · n is furnished by the Neumann boundary conditions prescribed on ΓN . 14) should be replaced by the algebraic relation ui = gi that follows from the Dirichlet boundary conditions.
It is worth mentioning that the formal order of approximation is not the sole indicator of accuracy and, in many cases, not even a particularly good one. The absolute values of the errors produced by two schemes of the same order may differ significantly, and a low-order scheme might perform better than a high-order one on a coarse mesh. Strictly speaking, a priori estimates based on Taylor series expansions are not applicable to discontinuous solutions, Also, the contribution of higher-order terms may become nonnegligible if the corresponding derivatives are too large.
A Guide to Numerical Methods for Transport Equations by Dmitri Kuzmin