• September 2025: the arXiv preprint of "Entropy stable finite difference methods via entropy correction artificial viscosity and knapsack limiting" by alumnus of the group Brian Christner is now online. This work extends two recently developed techniques for enforcing an entropy inequality (entropy correction artificial viscosity and knapsack limiting) to finite difference methods using a discrete graph-based artificial viscosity. The resulting scheme is high order accurate, and the knapsack limiting variant is able to preserve positivity for problems such as the LeBlanc shock tube and Woodward-Collela blast wave.
  
  

• September 2025: Jesse Chan gave a virtual talk in the Department of Mathematics "Modeling and Computation" seminar at the University of Arizona.

• August 2025: Jesse Chan gave a talk in the Babuska Forum at the Oden Institute.
  
  

• July 2025: the arXiv preprint of "Entropy Stable Nodal Discontinuous Galerkin Methods via Quadratic Knapsack Limiting" with graduate student Brian Christner is now online. This work generalizes our previous paper with Yimin Lin on enforcing an entropy inequality directly using subcell limiting. Lin computed optimal flux corrected transport (FCT) type limiting coefficients to enforce both invariant domain constraints and a cell entropy inequality. These optimal FCT coefficients are computed by solving a linear knapsack problem, whose solution can be computed using a simple and efficient deterministic algorithm.
  
  In this new preprint, Christner introduces a generalization of this approach using a quadratic knapsack problem, which significantly improves both the temporal accuracy and efficiency of the resulting discretization while retaining the efficiency and flexibility of the original knapsack problem. In particular, Christner shows that this new knapsack problem can be reduced to scalar root-finding problem with an efficient derivative formula and guaranteed convergence. We also demonstrate that the knapsack limiting entropy stable DG method is locally linearly stable, addressing an open theoretical issue with some high order entropy stable DG methods.
  
  

• July 2025: PhD student Ray Qu and Jesse Chan both gave talks at the ICOSAHOM 2025 and USNCCM 18 meetings. Jesse Chan's talk focused on recent work on entropy correction artificial viscosity for high order DG methods, while Ray Qu's talk was on recent work utilizing model reduction techniques for the stochastic finite volume method, a hyperbolicity-preserving method for uncertainty quantification for nonlinear conservation laws.
  
  

• July 2025: the arXiv preprint of "Model order reduction techniques for the stochastic finite volume method" with Ray Qu (Rice) and Svetlana Tokareva (LANL) is now available. The stochastic finite volume method (SFVM) developed by Tokareva, Abgrall, and others is an uncertainty quantification (UQ) technique for hyperbolic PDEs which preserves hyperbolicity. However, the resulting discretization can become very expensive for high dimensional stochastic parameter spaces.
  
  In this preprint, we utilize model reduction techniques (a reduced stochastic basis and hyper-reduction) to reduce the dimensionality of the stochastic part of the resulting system for 1D nonlinear conservation laws. In particular, we note that the reduced system does not require constructing the higher-dimensional SFVM system, and can be constructed purely by sampling deterministic 1D simulations.
  
  

• June 2025: Jesse Chan presented at the BIRS workshop "Structure Preserving Schemes for Complex Nonlinear Systems".

• May 2025: Jesse Chan and Sam van Fleet presented at the NSF Computational Mathematics meeting.

• April 2025: graduate student Brian Christner defended his MA thesis. 

• April 2025: PhD student Raymond Park defended his MA thesis. 

• April 2025: Jesse Chan gave a talk in the "Foundation Models of the Physical Universe" Symposium.

• April 2025: PhD student Raymond Park received an NSF Graduate Research Fellowship.

• March 2025: Jesse Chan gave a talk in the Applied Mathematics Seminar in the Dept. of Mathematics at the University of Utah.

• March 2025: Jesse Chan, Raven Johnson, Ray Qu, Raymond Park, and Brian Christner are all presenting at SIAM CSE 2025 in Fort Worth, TX. 

• Feb 2025: Jesse Chan gave a plenary talk on "Efficient Implementation of High Order Entropy Stable Methods for Computational Fluid Dynamics" at the Energy HPC Conference at Rice University.
  
  

• Feb 2025: the arXiv preprint of "Entropy stable reduced order modeling of nonlinear conservation laws using discontinuous Galerkin methods" with PhD student Ray Qu and Prof. Akil Narayan is now available. In this work, we generalize entropy stable reduced order models to high order DG methods, introducing a new condition to guarantee accuracy on non-uniform meshes. We also examine the performance of entropy stable ROMs and their robustness for predictive simulations.
  
  

• Jan 2025: the arXiv preprint of "An artificial viscosity approach to high order entropy stable discontinuous Galerkin methods" is now available. In this work, we show that it is possible to construct a semi-discretely entropy stable high order DG method on general elements using a simple local artificial viscosity coefficient computed from the local entropy violation and local entropy dissipation.
  
  

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