This paper discusses the discrete analogue of the gradient of a function and shows how discrete gradients can be used in the numerical integration of ordinary differential equations (ODEs). Given an ODE and one or more first integrals (i.e. constants of the motion) and/or Lyapunov functions, it is shown that the ODE can be rewritten as a 'linear-gradient system'. Discrete gradients are used to construct discrete approximations to the ODE which preserve the first integrals and Lyapunov functions exactly. The method applies to all Hamiltonian, Poisson and gradient systems, and also to many dissipative systems (those with a known first integral or Lyapunov function).

Systems with a first integral (i.e., constant of motion) or a Lyapunov function can be written as "linear-gradient systems"ẋ = L(x)∇V (x) for an appropriate matrix function L, with a generalization to several integrals or Lyapunov functions. The discrete-time analogue, ∆x/∆t = L∇V where ∇ is a "discrete gradient," preserves V as an integral or Lyapunov function, respectively.

Mimetic discretization methods for the numerical solution of continuum mechanics problems directly use vector calculus and differential forms identities for their derivation and analysis. Fully mimetic discretizations satisfy discrete analogs of the continuum theory results used to derive energy inequalities. Consequently, continuum arguments carry over and can be used to show that discrete problems are well-posed and discrete solutions converge. A fully mimetic discrete vector calculus on three dimensional tensor product grids is derived and its key properties proven. Opinions regarding the future of the field are stated.

No abstract

No abstract