Numerical Solution of PDEs

Numerical Methods for Two-Phase Mixture Models

In a series of papers, we developed efficient and accurate numerical methods for solving the equations of a two-material mixture model subject to an incompressibility condition on the volume-fraction averaged velocity. In the first three papers, the viscous dominated case was considered. The discrete momentum and incompressibility equations give rise to a saddle point problem which is solved using multi-grid (right-)preconditioned GMRES. Advection of volume fractions is discretized using a high-resolution finite-volume method. The overall scheme was shown to be robust, even during phase separation of the materials. The methods work well for both dilute and concentrated networks. A parallel version and a Cartesian-grid version to handle irregular geometry were also developed.

Grady B. Wright, Robert D. Guy, and Aaron L. Fogelson, An Efficient and Robust Method for Simulating Two-Phase Gel Dynamics, SIAM Journal on Scientific Computing, 30, 2008, 2535-2565.

Jian Du and Grady Wright and Aaron L. Fogelson, A Parallel Computational Method for Simulating Two-Phase Gel Dynamics, International Journal for Numerical Methods in Fluids, 60, 2009, 633-649.

Jian Du and Aaron L. Fogelson, A Cartesian Grid Method for Two-Phase Gel Dynamics on Irregular Domains, International Journal for Numerical Methods in Fluids, 67, (2011) 1799-1817.

The method was then extended to handle the equations retaining the inertia terms and also including a viscoelastic stress which evolves according to a modified Oldroyd-B type equation.

Grady B. Wright, Robert D. Guy, Jian Du, and Aaron L. Fogelson, A high-resolution finite-difference method for simulating two-fluid, viscoelastic gel dynamics, Journal of Non-Newtonian Fluid Mechanics, 166, (2011) 1137-1157.

In order to handle problems in which the network phase was present in only a portion of the computational domain, a regularization method was developed and validated

Jian Du, Robert D. Guy, Aaron L. Fogelson, Grady B. Wright, and James P. Keener, `An Interface-capturing Regularization Method for Solving the Equations for Two-fluid Mixtures, Communications in Computational Physics, 2013, 14(5), 1322-1346.

The time-dependent viscoelastic methods were combined with other methods to solve the equations of our two-phase continuum models of platelet aggregation.

The regularized methods for the viscous-dominated situation were recently combined with efficient methods to track ion transport by advection and electromigration, to enforce electroneutrality, and to include appropriate chemical potential gradients to capture short-range interaction and entropic forces between the materials. This method is being applied to our model of polyelectrolyte gel dynamics.

Jian Du, Bindi M. Nagda, Owen L. Lewis, Daniel B. Szyld, Aaron L. Fogelson, A computational framework for the swelling dynamics of mucin-like polyelectrolyte gels, Journal of Computational Physics, 2021, submitted.

As an alternative to the fully Eulerian two-phase mixture model just described, we develop an Eulerian-Lagrangian mixture model able to handle the dynamics of non-dilute gels. In this method, the network is represented in a Lagrangian manner and there is no need for regularization if the network occupies only a portion of the computational domain.

Victor Camacho, Aaron L. Fogelson, James P. Keener, Eulerian-Lagrangian Treatment of Non-dilute Two-phase Gels, SIAM Journal on Applied Mathematics, 2016, 76, 341-367.

The development of numerical methods for two-phase mixture models was supported by NSF Grant DMS-1160432 and NIGMS Grant 1R01GM131408.

Numerical Methods for Advection-Diffusion and Reaction-Diffusion Problems

In a series of papers, we describe our development of novel mesh-free Radial Basis Function (RBF) based finite-difference schemes for solving advection-diffusion equations in irregular domains and for solving reaction-diffusion equations on surfaces. The developments include automatic node generation and robust analysis-based specification of hyperviscosity for stabilization in polynomial-augmented RBF-FD discretizations of Laplacian and gradient operators. High order convergence is demonstrated.

Varun Shankar, Aaron L. Fogelson, Hyperviscosity-Based Stabilization for Radial Basis Function-Finite Difference (RBF-FD) Discretizations of Advection-Diffusion Equations, Journal of Computational Physics, 2018, 372, 616-639.

Varun Shankar, Robert M. Kirby, Aaron L. Fogelson, Robust Node Generation for Meshfree Discretizations on Irregular Domains and Surfaces, SIAM Journal on Scientific Computing, 2018, 40, A2584-2608.

Varun Shankar, Grady B. Wright, Robert M. Kirby, Aaron L. Fogelson, A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction-Diffusion Equations on Surfaces, Journal of Scientific Computing, 2016, 63, 745-768.

Varun Shankar, Grady B. Wright, Aaron L. Fogelson, Robert M. Kirby, A Radial Basis Function (RBF)-Finite Difference Method for the Simulation of Reaction-Diffusion Equations on Stationary Platelets within the Augmented Forcing Method, International Journal of Numerical Methods for Fluids, 2014, 75, 1-22.

This work was done in collaboration with Varun Shankar (Utah), Mike Kirby (Utah) and Grady Wright (Boise State U) and was supported by NSF Grant DMS-1521748.

High Order Interface Tracking Method

We developed and analyzed a highly effective high-order interface tracking algorithm for use in solving transport equations.

Qinghai Zhang, Aaron L. Fogelson, Fourth-order interface tracking by an improved polygonal area mapping (iPAM) method, SIAM Journal of Scientific Computing, 2014, 36, A2369-A2400.

Qinghai Zhang, Aaron L. Fogelson, MARS: An analytic framework of interface tracking via mapping and adjusting regular semi-algebraic sets, 2015, SIAM Journal on Numerical Analysis, 2016, 54, 530-560.

This work was done in collaboration with Qinghai Zhang (Zhejiang University) and was supported by NSF Grant DMS-1160432 and NIH Grant 1R01-GM090203.

Advection-Diffusion in Complex Time-dependent domains

We have explored multiple approaches to solving advection-diffusion-reaction equations in complex time-dependent domains with binding of fluid phase chemicals to the domain boundaries and chemistry on those boundaries. Both approaches use a semi-Lagrangian method for advection; one uses a cut-cell based finite-volume method for the reaction-diffusion terms while the other uses a high-order meshless RBF-FD method.

Aaron Barrett, Aaron L. Fogelson, Boyce E. Griffith, A Hybrid Semi-Lagrangian Cut Cell Method for Advection-Diffusion Problems with Robin Boundary Conditions in Moving Domains', Journal of Computational Physics, 2022, 449, 110805.

Varun Shankar, Grady B. Wright, Aaron L. Fogelson, An Efficient High-Order Meshless Method for Advection-Diffusion Equations on Time-Varying Irregular Domains, Journal of Computational Physics, 2021,445, 110633.

The cut-cell approach work was done in collaboration with Aaron Barrett (Utah) and Boyce Griffith (UNC). The RBF-FD approach was done in collaboration with Varun Shankar (Utah) and Grady Wright (Boise State). This work was partially supported by NHLBI Grant 1U01HL143336.U01HL1433361U01HL1