The authors consider the first-order time-dependent partial differential equation (PDE): \[ F(\partial_t u, u)=0, \] where \(F\) is a functional of \(\partial_t u, u\) and, possibly derivatives of \(u\) with respect to the other variables than time. The objective is to devise a method that combines the use of the coarse solver on the whole interval \((0,T)\) and the fine solver in parallel on each small interval \((T^n, T^{n+1})\) and allows to provide an approximation of \(u\) over the whole interval \((0,T)\) with the same accuracy as a global use of fine, one-step solver.
The authors introduce a slightly modified implementation, in comparison with the method of J.-L. Lions, Y. Maday and G. Turinici [C. R. Acad. Sci. Paris, Sér. I, Math. 332, 661-668 (2001; Zbl 0984.65085)], which turns out to be equivalent to linear problems, but gives better answers for nonlinear problems and allows to tackle non-differentiable problems. An example of such problems is provided by the Black-Scholes equation for an American put in financial mathematics. Numerical advantages of the proposed “parareal” time discretization are presented.
For the entire collection see [Zbl 0989.00043].