Towards information-optimal simulation of partial differential equations

Most simulation schemes for partial differential equations (PDEs) focus on minimizing a simple error norm of a discretized version of a field. This paper takes a fundamentally different approach; the discretized field is interpreted as data providing information about a real physical field that is unknown. This information is sought to be conserved by the scheme as the field evolves in time. Such an information theoretic approach to simulation was pursued before by information field dynamics (IFD). In this paper we work out the theory of IFD for nonlinear PDEs in a noiseless Gaussian approximation. The result is an action that can be minimized to obtain an information-optimal simulation scheme. It can be brought into a closed form using field operators to calculate the appearing Gaussian integrals. The resulting simulation schemes are tested numerically in two instances for the Burgers equation. Their accuracy surpasses finite-difference schemes on the same resolution. The IFD scheme, however, has to be correctly informed on the subgrid correlation structure. In certain limiting cases we recover well-known simulation schemes like spectral Fourier-Galerkin methods. We discuss implications of the approximations made.

Figure 1

Simulation of the Burgers equation using a Gaussian velocity profile as initial condition as represented by the gray line. The dotted line shows the reconstruction as it is obtained from the IFD formalism; the dashed line shows a finite-difference simulation with the same resolution. The solid line is a more exact simulation obtained by simulating with a finite-difference scheme in four-times higher spatial resolution.

Figure 2

Zoom-in into the simulation of the Burgers equation shown in Fig. 1. The dash-dotted line shows the reconstruction as it is obtained from the IFD formalism, but with double the spatial resolution. The other lines are the same as in Fig. 1.

Figure 3

Simulation of the Burgers equation using a Gaussian velocity profile as initial condition as represented by the gray line and a Fourier space response. The dotted line is a reconstruction that is obtained through simulating with the IFD scheme. This simulation scheme is equivalent to a Fourier-Galerkin scheme. The solid line is a high resolution simulation using a finite-difference scheme.