Research

Capturing structural properties, such as positivity, multi-scales, and asymptotic regimes, are essential in many applications of hyperbolic balance laws. However, when attempting to capture these properties on a discrete level, many numerical methods can return inaccurate, or even physically impossible results. One possible fix is to use a very fine spatial resolution, but this is not always worthwhile due to the extra amount of computation. One alternative is to use structure-preserving methods, in which the goal is to preserve the desired properties of a model in addition to being consistent with the system of balance laws. Currently, I am building upon the following existing ideas in structure-preserving schemes:

• Involution-Preserving Methods: Many applications require an exact preservation of the involution constraint within the hyperbolic system on a discrete level. One example of an involution constraint, or a time-independent differential equation in which the governing hyperbolic system must satisfy for all time, is the divergence-free condition of the magnetic field in the Maxwell or magnetohydrodynamic systems. If these constraints are not maintained exactly, large instabilities or inaccurate results may occur within numerical simulations. Methods that preserve these constraints are known as Involution-preserving methods.

• Well-Balanced Methods: Many applications result in solutions that are small-perturbations of steady-states. In many cases, these perturbations may be smaller than the truncation error of your numerical scheme. Well-balanced methods preserve steady-states exactly, and as a positive consequence, maintain the structures of small perturbations on a coarse spatial mesh.

• Asymptotic-Preserving Methods: In many applications, the system of interest contains some singular small parameter within, i.e., the fluxes and sources. Furthermore, these systems may show non-uniform behavior and may become very stiff in the limit as this small parameter approaches zero, thus making these simulations computationally expensive. One possible solution to these issues is an asymptotic-preserving method, in which the discretization of the continuous system remains consistent and stable regardless of the value taken for the underlying singular parameter; in other words, the CFL condition and the spatial discretization are independent of the underlying small parameter.