Sequential Data Assimilated Shape-Morphing Solutions (DA-SMS)

Partial Differential Equations (PDEs) are an integral tool for making forecasts. However various forms of errors can limit the predictive power of the associated model. Data assimilation seeks to mitigate these issues by incorporating observational data into the numerical approximation of the PDE. However, many existing methods are very computationally expensive. To address both of these issues, we use a non-linear approximation of the solution to the PDE that incorporates observational data into the approximation.

We have successfully applied DA-SMS to a few problems including the Nonlinear Schrödinger Equation, the Kuramoto-Sivashinsky equation (depicted below), and the 2D advection-diffusion equation.

Enforcing Conserved Quantities in Galerkin Truncation and Finite Volume Discretization

Many PDEs have associated conserved quantities such as total mass, energy, enstrophy etc. that the solution must satisfy. However, it is well known that these physical properties of the solution may not survive numerical approximation. We developed methodologies known as Galerkin Reduced-Order Nonlinear Solutions (G-RONS) and finite-volume reduced-order nonlinear solutions (FV-RONS) which ensures that these associated conserved quantities are respected by the numerical solution to the PDE.

We successfully applied this methodology to the Nonlinear Schrödinger equation and the shallow water equation. Our methodologies and results can be found in this paper:

Enforcing conserved quantities in Galerkin truncation and finite volume discretization, Z. T. Hilliard, M. FarazmandNonlinear Dynamics, vol. 112, pp. 14051-14069, 2024