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8: “Total” damping experienced by a disturbance crossing the domain, as a function J, the number of grid points across the domain. Here we have assumed that D ⁄ L = 2 . follows: • Check truncation error. Normally this is done by means of a Taylor series expansion. We are concerned with the lowest powers of the space and time grid-interval in the expansion of the independent variables. • Check linear stability for a simplified (linearized, constant coefficients) version of the equation. Von Neumann's method is often used here.

In general, “good” schemes have the following properties, among others: • High accuracy. • Stability. • Simplicity. • Computational economy. Later we will extend this list to include additional desirable properties. 10 Summary 41 offs. For example, a more accurate scheme is usually more complicated and expensive than a less accurate scheme. We have to ask whether the additional complexity and computational expense are justified by the increased accuracy. The answer depends on the particular application.

This is the exact solution. To work out the numerical approximation to this solution, we first choose ∆x and ∆t such that the grid points are the dots in the figure. 75). 6 Discretization error and convergence 27 We could increase the accuracy of the scheme by cutting ∆x and ∆t in half, that is, by adding the points denoted by small x's forming a denser grid. Notice that the domain of dependence does not change, no matter how refined or dense the grid is, so long as the c∆t c∆t ratio -------- remains the same.

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