By Dmitri Kuzmin
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Extra resources for A Guide to Numerical Methods for Transport Equations
The advantages of weakly imposed Dirichlet boundary conditions have been explored in [25, 176, 212, 293]. 13) is employed. At the initialization stage, the coefficients mi j , ci j , and di j need to be evaluated using numerical integration. In finite element codes, the matrix assembly process is fully automatic and involves transformations to a reference element on which the local basis functions are defined [76, 176]. In the case of a time-dependent velocity field, the convective part of the discrete operator K must be updated at each time step.
69) To secure convergence and keep the numerical solution positive, the splitting of the reactive term should be designed so that αi ≥ 0 and βi ≥ 0 for all i, see . 68) with nonnegative coefficients αi and βi . This negative-slope linearization technique was invented by Patankar  to preserve the sign of inherently positive variables. 68) can be found in his book. , for zeroth-order kinetics (radiative decay: r = −cu ⇒ α = c ≥ 0, β = 0), a combination of first- and second-order kinetics (population growth: r = c1 u − c2 u2 ⇒ α = c2 u ≥ 0, β = c1 u ≥ 0), and reactions of fractional order (for instance: r = −cu3/2 ⇒ α = cu1/2 ≥ 0, β = 0).
A possible remedy is to take a suitable high-order discretization and constrain it at the algebraic level so as to enforce desirable properties without losing too much accuracy. This is the approach that we will pursue and promote in this book. Most of the material presented in this text is not really new. Many excellent books and review articles have been written about numerical methods for convectiondiffusion equations and hyperbolic conservation laws. The main reasons that have led the author to retell the story are as follows: • • • • useful techniques are scattered over a vast body of literature and difficult to find; many algorithms are inherently explicit or require the use of structured meshes; texts overloaded with complex mathematical theory are unreadable to engineers; rigorous convergence proofs may disguise the fact that the method does not really work when applied to problems in which subgrid-scale effects are important.