Numerical Solution of Integral Equations by Michael A. Golberg
By Michael A. Golberg
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Example text
Assume that the nonnegative function z (ξ) = −q (g(ξ))g (ξ) attains its positive maximum z (ξ M ) only at the point ξ M ∈ [a, b]. Then ts = min ξ∈[a,b] 1 1 = . 4 The method of characteristics revisited 39 Since xs belongs to the characteristics x = q (g (ξ M )) t + ξ M , we find xs = q (g (ξ M )) + ξM . 2. The point (xs , ts ) has an interesting geometrical meaning. 43) admits an envelope and (xs , ts ) is the point on the envelope with minimum time coordinate. 43) with respect to ξ. Clearly, the envelope has not to be confused with the shock curve.
We conclude this minimal introduction to finite difference approximation of scalar conservation laws by addressing some considerations about stability, which clarify the behavior of the approximate solution compared to the original model and represent a necessary requirement to make sure that the numerical approximation converges to the exact solution u. We briefly address the CFL condition (from Courant-Friedrichs-Lewy). It requires that the speed at which the scheme propagates the initial state must not be smaller than the characteristic speed of the model, namely |a| for the case ut +aux = 0.
7) we deduce ct = Dcxx − vcx which constitutes our mathematical model. 3 4 Assuming we can take the derivative inside the integral. [q] = [mass] × [time]−1 . 1. Notice that if v and D were non constant, we would get an equation of the form ct = (Dcx )x − (vc)x . 8). We want to determine the evolution of the concentration c, by knowing its initial profile c (x, 0) = g (x) . 9) can be written in the form vcx + ct = ∇c · v =0, pointing out the orthogonality of ∇c and v. But ∇c is orthogonal to the level lines of c, along which c is constant.