General

The focus in my group is on developing efficient and robust numerical algorithms for the solution of general classes of partial differential equations based on novel numerical methods with provable properties. We are broadly interested in numerical methods ranging from finite-element methods to finite-difference and finite-volume methods. We are also working on approaches to dealing with complex geometry (e.g., unstructured meshes) and efficient computation (e.g., mesh adaptation).

PDEs and physical applications of interest are broad and constantly growing. For example, currently there are projects on the compressible Navier Stokes equations, multi-phase descriptions of fluids, linear and nonlinear wave equations, and reaction diffusion equations. Some current areas of application include aerospace applications and high intensity focused ultrasound. Below, a few past research contributions are highlighted.

The Summation-by-parts framework

Developing methods that have mathematically provable properties (e.g., stability and conservation) is a significant challenge for numerical analysts. This is particularly true if one considers high-order methods (these are well suited for modern super computers because of their dense compute kernels), complex geometry, nonlinear PDEs, and when coupling the solution of more than one PDE (e.g., fluid structure interactions). Fortunately, in the last decade, large strides have been made in generalizing the summation-by-parts (SBP) framework from its finite-difference origins to a flexible approach for the design and analysis of methods having provable properties (for linear and nonlinear PDEs) applicable to a broad set of existing and novel schemes. The strength of the SBP lays in the fact that it is a matrix analysis technology and can therefore be used to analyze the actual schemes that are implemented on a computer regardless of the origin of the scheme (e.g., finite-element, finite difference, or finite volume to name a few). We have worked extensively in this area pushing the boundaries of applicability of the SBP framework, where currently, it is capable of analyzing existing and novel methods derived from diverse discretization techniques such as finite elements, flux reconstruction, finite difference, unstructured methods etc. (For tensor product operators see paper1, paper2, paper3 paper4, paper5, paper6; for multidimensional operators see paper1, paper2, paper3, paper4; for discontinuous Galerkin schemes see paper1, paper2 ; for flux reconstruction schemes see paper1, paper2). Moreover, it gives a tractable approach for examining existing schemes and determining minimally invasive (design-order) changes that result in provable properties. The figure below presents the state of the art in SBP operators. What we see in the figure is that general shapes can be used for elements describing the mesh and that very general constructions with nodes or modes can be captured by the SBP framework.

Entropy stable discretizations of the compressible Euler and Navier-Stokes equations

Provable stability is fundamental to developing useful numerical algorithms. Without provable stability a simulation can diverge and cause a simulation to fail or worse yet provide results that are highly inaccurate (imagine a weak instability that does not cause a code to blow up during simulation but causes the numerical solution to deviate wildly from the true solution). Moreover, provable stability is a key enabling technology for high-order methods which in turn are one possible path to efficient use of high-performance computing resources. However, developing provably stable high-order methods is challenging. For a certain class of PDEs, at the continuous level, stability can be proven using entropy analysis, for example the compressible Euler and Navier-Stokes equations, but the concept is more broadly applicable and in a lose sense is the nonlinear extension of energy analysis used for linear PDEs. For example in the figures below

Entropy stable h/p adaptation

Once a scheme is provably stable, then it is possible to turn to questions of efficiency. For multi scale problems, i.e., where there is a large separation in the scale of relevant physical processes, the required degrees of freedom needed to obtain certain error level can be significantly reduced by using mesh adaptation. In this regard, we have looked at two technologies, p-adaptation, where the number of degrees of freedom in a given element are increased or decreased and h-adaptation, where the mesh is locally refined or coarsened (see paper1 paper2, paper3). The figures below show the application of p refinement to first a delta wing and second to the flow around a cylinder with degree low to high from ordered as p=2 (blue), p=3 (green), p=4 (orange), and p=5 (red) (see paper4).

a) Delta-wing p refinement                                                                                                                          b) Cylinder p refinement

c) Average flow filed: streamlines (left) and mean axial velocity (right)                                                          d) Isosurfaces of Q-criterion

Arbitrary Lagrangian Eulerian framework for r adaptation

For problems that undergo large displacements and deformations (e.g., fluid-structure interactions) a common approach is to employee the arbitrary Lagrangian-Eulerian (ALE) framework. In this approach, the physical equations are mapped into a computational grid that can move and deform independently of the fluid flow. Besides problems with deformations the ALE approach can be used to optimize the placement of nodes without increasing the degrees of freedom of the underlying computation, and is therefore also an attractive approach mesh adaptation. However, without care it is easy to destroy the stability and conservation properties of the underlying method. In this regard, we have looked at extending the entropy-stable discretization schemes discussed above for the compressible Navier-Stokes equations in order to enable the ALE approach while retaining provable stability and conservation (see paper1). The figure below shows a the density profile for the convecting vortex problem for the compressible Navier-Stokes equations for a stationary rectilinear mesh and a moving mesh computed using the ALE approach (see paper1).

a) Stationary mesh                                                              b) moving mesh