B-Methods for the Numerical Solution of Evolution Problems with Blow-Up Solutions Part I: Variation of the Constant

In the last two decades, the field of geometric numerical integration and structure-preserving algorithms has focused on the design of numerical methods that preserve properties of Hamiltonian systems, evolution problems on manifolds, and problems with highly oscillatory solutions. In this paper, we show that a different geometric property, namely, the blow up of solutions in finite time, can also be taken into account in the numerical integrator, giving rise to geometric methods we call B-methods. We give a first systematic approach for deriving such methods for scalar and systems of semi- and quasi-linear parabolic and hyperbolic partial differential equations. We show both analytically and numerically that B-methods have substantially better approximation properties than standard numerical integrators as the solution approaches the blow-up time.