By Jayathi Y. Murthy
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Extra resources for Numerical Methods in Heat, Mass, and Momentum Transfer (draft)
So we are assured that under-relaxation is only a change in the path to solution, and not in the discretization itself. Thus, both under-relaxed and un-underrelaxed equations yield the same final solution. ✧ ✧ 2. Though over-relaxation (α 1) is a possibility, we will for the most part be using α 1. With α 1, we are assured that a P α ∑nb anb . This allows us to satisfy the Scarborough criterion. ✳ ✳ ✢ 3. The optimum value of α depends strongly on the nature of the system of equations we are solving, on how strong the non-linearities are, on grid size and so on.
16 may be derived for all cells in the domain, yielding a set of algebraic equations, as before; these may be solved using a variety of direct or iterative methods. We note the following about the discretization process. 1. The process starts with the statement of conservation over the cell. We then find cell values of φ which satisfy this conservation statement. Thus conservation is guaranteed for each cell, regardless of mesh size. 32 2. Conservation does not guarantee accuracy, however. The solution for φ may be inaccurate, but conservative.
1 Accuracy Accuracy refers to the correctness of a numerical solution when compared to an exact solution. In most cases, we do not know the exact solution. It is therefore more useful to talk of the truncation error of a discretization method. 5, the terms that are neglected are of O ∆x 2 . Thus, if we refine the mesh, we expect the truncation error to decrease as ∆x 2 . If we double the x-direction mesh, we expect the truncation error to decrease by a factor of four. The truncation error of a discretization scheme is the largest truncation error of each of the individual terms in the equation being discretized.