Structure-preserving regularization and stochastic forcing for nonlinear hyperbolic PDEs

Structure-preserving regularization and stochastic forcing for nonlinear hyperbolic PDEs (SPReSto)



Nonlinear hyperbolic systems partial differential equations (PDEs) play a role in many natural processes. Important topics such as weather prediction and turbulence rely heavily on such systems of equations. These equations do not only appear in natural processes, many industrial applications relating to water, energy and chemistry lean on accurate predictions of complex flows. The challenge is thus to predict flow solutions in an accurate and computationally feasible way. 

Systems of nonlinear hyperbolic PDEs usually display a wide range of dynamic scales. Resolving all scales in a prediction is required in order to gain a good grasp of the dynamics displayed in such systems. However, this is infeasible due to resolution issues; not all details can be handled with the current available computing resources and algorithms.

This problem is easily averted by cleverly using coarsened models. These models require fewer computing resources and, if chosen correctly, they provide reasonable solutions to problems that do not hinge on the finest details of the solution. It is one of the main aims of the project to find out how to systematically derive coarsened models that agree with the underlying dynamics, how to create structure-preserving numerical methods, and how to control the simulation error that appears when using these models and methods.

The approach to obtain accurate numerical models for coarsened solution predictions leans on deterministic regularization of the underlying equations, accompanied by stochastic forcing to the model. This stochastic forcing becomes especially important at coarse spatial resolutions. When derived from spatial correlations (Holm, 2015), the stochastic forcing represents dynamically relevant scales that are not explicitly available. At higher spatial resolutions, the main dynamics of the system of equations are captured by the deterministic regularization.