By Dmitri Kuzmin
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Extra resources for A Guide to Numerical Methods for Transport Equations
12) and the velocity field is not divergence-free. In this case, the governing equation contains a zeroth-order term of the form (∇ · v)u, so the rule is not applicable to the discrete convection operator. If we force the numerical solution to behave in a certain way, then our intention is to mimic some qualitative properties of the exact solution. The basic rules should not be used blindly in situations when the underlying assumptions do not hold. 3 Positive Coefficients The third basic rule is: if convection and diffusion are the only processes to be simulated, the nodal value uin+1 should not decrease as result of increasing any other nodal value that appears in the discretized equation for node i.
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.
The discretization error εh∆ t depends on the mesh size h and time step ∆ t. It can be estimated using Taylor series expansions or, in the case of finite element methods, sophisticated tools of functional analysis. Roundoff errors due to the finite precision of computer arithmetics are usually much smaller than εh∆ t , whereas iteration errors depend on the prescribed tolerances and stopping criteria for linear solvers. A properly designed numerical scheme must be sufficiently accurate and converge to the exact solution of the differential equation as the mesh size h and time step ∆ t are refined.
A Guide to Numerical Methods for Transport Equations by Dmitri Kuzmin