| Contents |
| We propose the higher order extension of numerical methods that preserve the energy-diminishing feature of a system with Lyapunov function. Therefore we focus on ODEs with a Lyapunov function, so that they can be rewritten in the form of a linear gradient, i.e, the right hand side of the ODE consists of the product of a negative-definite matrix and the gradient of the Lyapunov function. Then, the formal construction of discrete gradient methods is straightforward, since they amount to replacing the negative definite matrix with an approximation, and the gradient with a discrete gradient. There is considerable freedom in the choice of these parameters. The technique used in this contribution results from composing a first order discrete gradient method together with its adjoint, yielding a second order method. The basis first order method is based upon the component-wise discrete gradient, whereas the adjoint method is computed in a similar vein, but the order of components is reversed. Finally the proposed method is validated by numerical experiments, showing the main features of the method: it preserves the Lyapunov function, it approximates the continuous system up to second order, and it can be computed explicitly in particular examples. |
|