For semilinear strictly hyperbolic systems of the form ∂u/∂t + λ(x,t) ∂u/∂x = f(x,t,u) locally bounded solutions u are uniquely determined by Cauchy data. The problem of describing the location and strengths of all the singularities of u has been studied extensively. The two general principles for the propagation of singularities in hyperbolic partial differential equations in one space variable age: (1) When two singularity-bearing characteristics cross, the point of intersection becomes, in general, a source of new singularities travelling on all forward characteristics from that point. That is, u may be singular on an out-going characteristic from such an intersection point, even though u is smooth on the corresponding incoming characteristic. (2) If two incoming singularities have strengths n1 and n2, then the new outgoing singularity will, in general, have strength n1 + n2 + 2, where the strength is defined as the number of continuous derivatives.
First we study propagation of singularities for semilinear strictly hyperbolic systems for distribution initial data. Since the analytical study is difficult, we study the solutions numerically and demonstrate the behavior predicted by the sum law above. The key to our numerical study is that distribution data are idealizations of classical initial data. For example, the delta function is an idealization of a high localized peak.
Next we study propagation of singularities for semilinear strictly hyperbolic systems for piecewise smooth initial data. The accuracy of numerical approximations to piecewise smooth solutions is greatly influenced by the presence of singularities in the solution. In the region where the analytic solution is known to be smooth, high-order numerical approximations can lose accuracy if coupling is present. We analyze this phenomenon for linear dissipative schemes and identify the necessary techniques to obtain an optimal rate of convergence in this region.