By Dmitri Kuzmin
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Additional resources for A Guide to Numerical Methods for Transport Equations
People from this group typically favor implicit algorithms, finite element discretizations, and unstructured meshes. Inviscid flows are governed by hyperbolic conservation laws, and the traditional solution strategy relies on explicit finite difference or finite volume discretizations. Of course, there are many notable exceptions to this rule, and some diffusion of ideas takes place at the interface between the two worlds. In our experience, a lot can be gained by looking at what is going on both sides of this interface.
Typical side effects are strong numerical diffusion and/or spurious oscillations (wiggles, ripples). Arguably, an overly diffusive scheme is the lesser of the two evils if it is guaranteed to be bounded. ” We take the liberty to formulate another rule: a good algorithm must contain just as much numerical dissipation as is necessary to avoid nonphysical artifacts. Moreover, the mesh should be refined adaptively in troublesome regions where small-scale effects are present. In most cases, the bizarre behavior of numerical solutions is due the fact that • the model is dominated by convection, anisotropic diffusion, or stiff reaction; • discontinuities, steep gradients, and/or interior/boundary layers are involved; • the employed numerical scheme violates at least one of the four basic rules.
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.