Overset Grid Methods

Overset grid methods have been used to develop efficient and high-order accurate finite difference and finite volume methods for problems in complex geometry. They are especially useful for problems with moving and deforming boundaries and interfaces. See overtureFramework.org for more information and a list of publications.

Adaptive Variational Multiscale Methods

The variational multiscale (VMS) method has been used to develop finite element formulations for advection-dominated problems. In the VMS method, a projection is carried out to achieve a scale decomposition or separation. Such a scale separation has been used to model the subgrid-scale stresses in a turbulent flow to perform large-eddy simulations (LES) for complex-geometry problems involving unstructured meshes. This has been combined with a local form of the variational Germano identity to dynamically compute the unknown model parameters (e.g., eddy-viscosity) and apply dynamic LES for inhomogeneous turbulent flows. In addition, scale-separation due to the VMS method has been used to develop an explicit a-posteriori error estimator and drive mesh adaptation for complex problems including those with evolving geometries.

Summation-by-parts Methods

Summation-by-parts (SBP) operators mimic integration by parts at the discrete level; that is, at the level of numerical quadrature.  They were originally proposed by Kreiss and Scherer (https://doi.org/10.1016/B978-0-12-208350-1.50012-1) in order to construct stable discretizations of linear hyperbolic partial differential equations.

For many years, SBP methods remained somewhat obscure, but interest in these methods grew dramatically when Fisher and Carpenter — see, for example, https://dx.doi.org/10.1016/j.jcp.2013.06.014 — showed that SBP operators could be used to construct high-order, entropy-stable discretizations for hyperbolic conservation laws, such as the Euler equations of gas dynamics.  Fisher and Carpenter's work was significant, because the widespread adoption of high-order discretizations for compressible flows has been hindered, in part, by their “fragility” and lack of robustness.

SCOREC researchers have made fundamental contributions to the theory and implementation of SBP methods, which have helped extend these methods beyond conventional, multi-block finite-difference methods.  Some of our contributions to the field are summarized below.

• Superconvergence of functionals based on SBP operators (http://dx.doi.org/10.1137/100790987).
• A definition of multi-dimensional SBP operators suitable for triangular and tetrahedral elements (http://dx.doi.org/10.1137/15m1038360) and curved elements (http://dx.doi.org/10.1016/j.jcp.2017.12.015).
• Entropy-stable, discontinuous Galerkin-difference discretizations based on the SBP property (https://doi.org/10.1016/j.cam.2022.114885).
• SBP operators defined on point clouds over complex domains (https://doi.org/10.1016/j.jcp.2025.113940).

Researchers interested in experimenting with multi-dimensional SBP discretizations can take advantage of the Julia package SummationByParts.jl, which is available as a registered package through the Julia package ecosystem.

Immersed Finite Element Methods

To enable accurate and efficient multiphysics simulations the immersed finite element method (IFEM) has been developed. IFEM is based on the immersed boundary method, in which finite element approaches are used to handle complex fluid–structure interactions and other coupled phenomena without requiring conformal mesh generation. Flexible overlapping domain and interface treatments and robust numerical schemes have been incorporated to improve stability, scalability, and applicability across a wide range of engineering and biomedical problems. To support modular implementation and flexible coupling of multiple physics, the OpenIFEM software (https://github.com/OpenIFEM/OpenIFEM) has been developed and maintained as an open-source platform. OpenIFEM is distributed under the Apache 2.0 License and uses CMake to configure and build based on user-specified applications, which further provides flexibility and ease of deployment.

Partitioned Methods for Fluid Structure Interaction (FSI)

FSI, a particular instance of coupled multi-physics systems, is commonly encountered in many fields of science and engineering, and has many important applications such as wind power, blast-structure interaction, biological flows, and micro-fluidic devices. Traditional partitioned FSI schemes do not take into account the strong coupling between the fluid and solid, and can therefore exhibit an instability whereby an over-reaction of a light solid to an applied force from the fluid leads in turn to an even larger reaction from the fluid and so on. The focus of our work has been addressing these difficulties, and we have shown the construction of stable and accurate algorithms that overcome the added-mass instability for a variety of FSI regimes including those with both compressible and incompressible flows. The essence of the work is to use detailed analysis of the continuous problem to reveal exact interface conditions which, after discretization, yield superior stability and accuracy properties. We generically term schemes built in this way as Added Mass Partitioned (AMP). 

AMP scheme: Conjugate Heat Transfer (CHT)

Similar coupling techniques are developed for other multi-domain problems. In the examples below we show just conjugate heat transfer in a mock reactor, in the middle we show fully coupled heat transfer and incompressible flow around heated letters RPI, and at bottom we show electromagnetic wave propagation through dispersive metals.