We present a unified scheme which, by combining backward differentiation formulae and variational optimisation theory, significantly extends the scope of both concepts and their application to optimisation problems wherein the constituent dynamics consist of widely disparate scales. Our approach is based on the time-integration of a series of matrix Riccati expressions which satisfy a Hamilton–...

Abstract. The Modified Phase Field Crystal (MPFC) equation, a generalized damped wave equation for which the usual Phase Field Crystal (PFC) equation is a special case, is analyzed in detail in three dimensions. A time-discrete numerical scheme, based on a convex splitting for the functional energy, is utilized to construct an approximate solution, which is then shown to converge to a solution ...

Convergence of a full discretisation method is studied for a class of nonlinear second-order in time evolution equations, where the nonlinear operator acting on the first-order time derivative of the solution is supposed to be hemicontinuous, monotone, coercive and to satisfy a certain growth condition, and the operator acting on the solution is assumed to be linear, bounded, symmetric, and str...

The time discretisation of the initial-value problem for a first-order evolution equation by the two-step backward differentiation formula (BDF) on a uniform grid is analysed. The evolution equation is governed by a time-dependent monotone operator that might be perturbed by a time-dependent strongly continuous operator. Well-posedness of the numerical scheme, a priori estimates, convergence of...

When differential-algebraic equations of index 3 or higher are solved with backward differentiation formulas, the solution can have gross errors in the first few steps, even if the initial values are equal to the exact solution and even if the stepsize is kept constant. This raises the question of what are consistent initial values for the difference equations. Here we study how to change the e...

in this paper, we consider an implicit block backwarddifferentiation formula (bbdf) for solving volterraintegro-differential equations (vides). the approach given in thispaper leads to numerical methods for solving vides which avoid theneed for special starting procedures. convergence order and linearstability properties of the methods are analyzed. also, methods withextensive stability region ...