Note on Hyperbolic Problems

These are equations where advection strongly dominates diffusion. They occur when the unknown variable is primarily influenced by, say, a flow that carries it. They also occur in continuous age-structured models, where the role of the flow is played by the unidirectional transition from a younger subgroup to the older one. In either case, the underlying equation is [1], where D = 0, or its generalization for the case of two partial dimensions or several coupled variables. (Therefore, in what follows we refer only to the case D = 0 without mentioning this every time.) Each of such equations must have exactly one boundary condition specified at the endpoint where the flow emanates. For example, when solved over an interval

[0, X] with some X> 0, [1] with g> 0 describes advection toward increasing values of x. Therefore, such an equation must have one boundary condition at x = 0. For g < 0, the flow is in the opposite direction, and hence the boundary condition must be specified at x = X.

For g> 0, the following simple upwind method can be used:

either in their profile, the slope, or the curvature, it is more appropriate to use the method of characteristics. The idea of this method is founded in the fact that in hyperbolic problems, the solutions propagate along certain lines in space and time called characteristics. Then the evolution along a characteristic is governed by an ODE rather than a PDE. The method of characteristics is easy to use either for a single equation [1] or when the coefficients gs in the system of coupled such equations do not depend on x and lis (but may depend on t).

while for g < 0 one needs to replace the numerator on the RHS with Um — Um— 1. These methods have accuracy O(t) + 0(h). (For either sign of g, a method of accuracy O(t ) + 0(h ) results when one uses the central difference of the form [14] for both the temporal and spatial discretizations.) Note that using a wrong spatial discretization for a given temporal one (or vice versa) will result in an unstable, and therefore useless, method. The reason for this is similar to that why certain methods work well for ODEs with dissipation but do not work for conservative ODEs, and vice versa; see Numerical Methods for Local Models for more details. Upwind, downwind, and central-difference schemes of higher accuracy can be found in textbooks. All such methods are adequate when the solution does not have a sharp propagation front. For solutions with abrupt changes

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