Fixed mesh approximation of ordinary differential equations with impulses

Abstract

When trains of impulse controls are present on the right-hand side of a system of ordinary differential equations, the solution is no longer smooth and contains jumps which can accumulate at several points in the time interval. In technological and physical systems the sum of the absolute value of all the impulses is finite and hence the total variation of the solution is finite. So the solution at best belongs to the space BV of vector functions with bounded variation. Unless variable node methods are used, the loss of smoothness of the solution would a priori make higher-order methods over a fixed mesh inactractive. Indeed in general the order of \(L^2\)-convergence is \(h^{1/2}\) and the nodal rate is \(h\). However in the linear case \(L^2\)-convergence rate remains \(h^{1/2}\) but the nodal rate can go up to \(h^{K+2}\) by using one-step or multistep scheme with a nodal rate up to \(h^{2K+2}\) when the solution belongs to \(H^{K+1}\). Proofs are given of error estimates and several numerical experiments confirm the optimality of the estimates.