By Dmitri Kuzmin
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Extra resources for A Guide to Numerical Methods for Transport Equations
9) 44 2 Finite Element Approximations For the time being, the nodal values v j are assumed to be known. In real-life applications, they are computed numerically from a momentum balance equation. 6). Instead of introducing separate trial solutions for each variable, it is possible to interpolate the nodal values of a function that depends on a group of variables. This approximation technique is known as the group finite element formulation [109, 110, 111]. It eliminates the need for dealing with products of basis functions and leads to simpler discretized equations, which results in significant savings especially in the case of highly nonlinear problems.
2) where n denotes the unit outward normal. The appropriate choice of initial/boundary conditions depends on the problem at hand and on the available information. In what follows, we consider continuous (linear or multilinear) finite element discretizations and illustrate their relationship to finite volumes. 1), we derive the coefficient matrices and examine their properties. We revisit the aspects of conservation, upwinding, and artificial diffusion in the FEM context. Also, we split the contributions of discrete convection and diffusion operators into skew-symmetric numerical fluxes that represent the rate of bilateral mass transfer between two neighboring nodes.
High-order schemes perform well for smooth data but may produce unbounded solutions in the neighborhood of steep fronts. A conservative and bounded convergent scheme of high order is rather difficult to design and expensive to run but the results are usually rewarding. Indeed, if the total amount of the conserved quantity is correct, its distribution is accurate, and all relevant upper/lower bounds are satisfied, then the numerical solution must be very close to the exact one. Boundedness implies strong stability, so a consistent and bounded scheme is convergent.
A Guide to Numerical Methods for Transport Equations by Dmitri Kuzmin