By J. Matteson
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Extra info for A Coll. of Diophantine Probs. With Solns. [Classic Reprint]
4. We note the similarity of under-relaxation to time-stepping. The initial guess acts as the initial condition. The terms a P α and 1 α α a p φP represent the effect of the unsteady terms. 7 Discussion At this point, our discretized equation set is ready for solution. We have seen that our diffusion equation can be discretized to yield coefficients that guarantee physical plausibility for the orthogonal meshes we have considered here. This property is very useful in a numerical scheme. However, we will see in the next chapter that it cannot usually be obtained when meshes are non-orthogonal or unstructured.
E. all paths to solution) which arrive at a solution will give us the same solution for the same set of discrete equations. For non-linear problems, we do not have this guarantee, and the answer we get may depend on factors like the initial guess, and the actual path to solution. Though this is an important issue in computing fluid flows, we will not address it here. Solution methods may be broadly classified as direct or iterative. We consider each briefly below. 18) ✂✸✷ ✓ ✓✰✯✱✯✱✯ ✹ T is a vector consisting of the discrete where A is the coefficient matrix, φ φ 1 φ2 values of φ , and B is the vector resulting from the source terms.
121) ☎ ✠ This is always guaranteed since all terms are positive. 56. 125) 4Γ A similar analysis may also be done to obtain the criterion for three-dimensional situations. It is also possible to show, using a similar analysis, that the fully-implicit and Crank-Nicholson schemes are unconditionally stable. Though the von Neumann stability analysis and our heuristic requirement of positive coefficients have yielded the same criterion in the case of the explicit scheme, we should not conclude that the two will always yield identical results.