arXiv subjects

Jesse Chan

Publications and source records attributed to Jesse Chan.

At least 19 recordsLinked to original sources

arXiv2026

Entropy correction artificial viscosity for high order DG methods using multiple artificial viscosities

Entropy stable discontinuous Galerkin (DG) methods display improved robustness for problems with shocks, turbulence, and under-resolved features by enforcing an entropy inequality. Such methods have traditionally relied on entropy conservative (EC) fluxes that are computationally expensive to evaluate. An alternative approach for enforcing an entropy inequality is through a minimally dissipative ``entropy correction" artificial viscosity. We review how to construct such an artificial viscosity formulation and extend this approach to multiple types of viscosity (e.g., viscosity and thermal diffusivity). We determine simple analytical expressions for optimal viscosity parameters. We compare this to the case of a single monolithic viscosity parameter for different 1D and 2D problems, and show that the proposed method allows users to more precisely target specific physical phenomena while retaining robustness for general problem settings.

math.NA↗

arXiv2026

Volume Term Adaptivity for Discontinuous Galerkin Schemes

We introduce the concept of volume term adaptivity for high-order discontinuous Galerkin (DG) schemes solving time-dependent partial differential equations. Termed v-adaptivity, we present a novel general approach that exchanges the discretization of the volume contribution of the DG scheme at every Runge-Kutta stage based on suitable indicators. Depending on whether robustness or efficiency is the main concern, different adaptation strategies can be chosen. Precisely, the weak form volume term discretization is used instead of the entropy-conserving flux-differencing volume integral whenever the former produces more entropy than the latter, resulting in an entropy-stable scheme. Conversely, if increasing the efficiency is the main objective, the weak form volume integral may be employed as long as it does not increase entropy beyond a certain threshold or cause instabilities. Thus, depending on the choice of the indicator, the v-adaptive DG scheme improves robustness, efficiency and approximation quality compared to schemes with a uniform volume term discretization. We thoroughly verify the accuracy, linear stability, and entropy-admissibility of the v-adaptive DG scheme before applying it to various compressible flow problems in two and three dimensions.

math.NA↗

arXiv2026

On the choice of viscous discontinuous Galerkin discretization for entropy correction artificial viscosity methods

Entropy correction artificial viscosity (ECAV) is an approach for enforcing a semi-discrete entropy inequality through an entropy dissipative correction term. The resulting method can be implemented as an artificial viscosity with an extremely small viscosity coefficient. In this work, we analyze ECAV when the artificial viscosity is discretized using a local discontinuous Galerkin (LDG) method. We prove an $O(h)$ upper bound on the ECAV coefficient, indicating that ECAV does not result in a restrictive time-step condition. We additionally show that ECAV is contact preserving, and compare ECAV to traditional shock capturing artificial viscosity methods.

math.NA↗

arXiv2025

Efficient and Robust Carath\'{e}odory-Steinitz Pruning of Positive Discrete Measures

In many applications, one seeks to approximate integration against a positive measure of interest by a positive discrete measure: a numerical quadrature rule with positive weights. One common desired discretization property is moment preservation over a finite dimensional function space, e.g., bounded-degree polynomials. Carath\'{e}odory's theorem asserts that if there is any finitely supported quadrature rule with more nodes than the dimension of the given function space, one can form a smaller (and hence more efficient) positive, nested, quadrature rule that preserves the moments of the original rule. We describe an efficient streaming procedure for Carath\'{e}odory-Steinitz pruning, a numerical procedure that implements Carath\'{e}odory's theorem for this measure compression. The new algorithm makes use of Givens rotations and on-demand storage of arrays to successfully prune very large rules whose storage complexity only depends on the dimension of the function space. This approach improves on a naive implementation of Carath\'{e}odory-Steinitz pruning whose runtime and storage complexity are quadratic and linear, respectively, in the size of the original measure. We additionally prove mathematical stability properties of our method with respect to a set of admissible, total-variation perturbations of the original measure. Our method is compared to two alternative approaches with larger storage requirements: non-negative least squares and linear programming, and we demonstrate comparable runtimes, with improved stability and storage robustness. Finally, we demonstrate practical usage of this algorithm to generate quadrature for discontinuous Galerkin finite element simulations on cut-cell meshes.

math.NA↗

arXiv2025

Entropy stable finite difference methods via entropy correction artificial viscosity and knapsack limiting

Entropy stable methods have become increasingly popular in the field of computational fluid dynamics. They often work by satisfying some form of a discrete entropy inequality: a discrete form of the 2nd law of thermodynamics. Schemes which satisfy a (semi-)discrete entropy inequality typically behave much more robustly, and do so in a way that is hyperparameter free. Recently, a new strategy was introduced to construct entropy stable discontinuous Galerkin methods: knapsack limiting, which blends together a low order, positivity preserving, and entropy stable scheme with a high order accurate scheme, in order to produce a high order accurate, entropy stable, and positivity preserving scheme. Another recent strategy introduces an entropy correction artificial viscosity into a high order scheme, aiming to satisfy a cell entropy inequality. In this work, we introduce the techniques of knapsack limiting and artificial viscosity for finite difference discretizations. The proposed schemes preserve high order accuracy in sufficiently smooth conditions, are entropy stable, and are hyperparameter free. Moreover, the proposed knapsack limiting scheme provably preserves positivity for the compressible Euler and Navier-Stokes equations. Both schemes achieve this goal without significant performance tradeoffs compared to state of the art stabilized schemes.

math.NA↗

arXiv2025

Entropy Stable Nodal Discontinuous Galerkin Methods via Quadratic Knapsack Limiting

Lin, Chan (High order entropy stable discontinuous Galerkin spectral element methods through subcell limiting, 2024) enforces a cell entropy inequality for nodal discontinuous Galerkin methods by combining flux corrected transport (FCT)-type limiting and a knapsack solver, which determines optimal limiting coefficients that result in a semi-discrete cell entropy inequality while preserving nodal bounds. In this work, we provide a slight modification of this approach, where we utilize a quadratic knapsack problem instead of a standard linear knapsack problem. We prove that this quadratic knapsack problem can be reduced to efficient scalar root-finding. Numerical results demonstrate that the proposed quadratic knapsack limiting strategy is efficient and results in a semi-discretization with improved regularity in time compared with linear knapsack limiting, while resulting in fewer adaptive timesteps in shock-type problems.

math.NA↗

arXiv2025

Model Order Reduction Techniques for the Stochastic Finite Volume Method

The stochastic finite volume method (SFV method) is a high-order accurate method for uncertainty quantification (UQ) in hyperbolic conservation laws. However, the computational cost of SFV method increases for high-dimensional stochastic parameter spaces due to the curse of dimensionality. To address this challenge, we incorporate interpolation-based reduced order model (ROM) techniques that reduce the cost of computing stochastic integrals in the SFV method. Further efficiency gains are achieved through hyper-reduction with a QR factorization-based discrete empirical interpolation method (Q-DEIM). Numerical experiments suggest that this approach can lower both computational cost and memory requirements for high-dimensional stochastic parameter spaces.

math.NA↗

arXiv2025

Entropy stable reduced order modeling of nonlinear conservation laws using discontinuous Galerkin methods

Reduced order models (ROMs) are inexpensive surrogate models that reduce costs associated with many-query scenarios. Current methods for constructing entropy stable ROMs for nonlinear conservation laws utilize full order models (FOMs) based on finite volume methods (FVMs). This work describes how to generalize the construction of entropy stable ROMs from FVM FOMs to high order discontinuous Galerkin (DG) FOMs. Significant innovations of our work include the introduction of a new "test basis" which significantly improves accuracy for DG FOMs, a dimension-by-dimension hyper-reduction strategy, and a simplification of the boundary hyper-reduction step based on "Carath\'eodory pruning".

math.NA↗

arXiv2025

An artificial viscosity approach to high order entropy stable discontinuous Galerkin methods

Entropy stable discontinuous Galerkin (DG) methods improve the robustness of high order DG simulations of nonlinear conservation laws. These methods yield a semi-discrete entropy inequality, and rely on an algebraic flux differencing formulation which involves both summation-by-parts (SBP) discretization matrices and entropy conservative two-point finite volume fluxes. However, explicit expressions for such two-point finite volume fluxes may not be available for all systems, or may be computationally expensive to compute. This paper proposes an alternative approach to constructing entropy stable DG methods using an entropy correction artificial viscosity, where the artificial viscosity coefficient is determined based on the local violation of a cell entropy inequality and the local entropy dissipation. The resulting method is a modification of the entropy correction introduced by Abgrall, Offner, and Ranocha (2022) in "Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization", and recovers the same global semi-discrete entropy inequality that is satisfied by entropy stable flux differencing DG methods. The entropy correction artificial viscosity coefficients are parameter-free and locally computable over each cell, and the resulting artificial viscosity preserves both high order accuracy and a hyperbolic maximum stable time-step size under explicit time-stepping.

math.NA↗

arXiv2024

An Entropy Stable High-Order Discontinuous Galerkin Method on Cut Meshes

High-order entropy stable summation-by-parts (SBP) schemes are a class of robust and accurate numerical methods for hyperbolic conservation laws that are numerically stable at arbitrary order without the need for artificial stabilization. While SBP schemes are well-established on simplicial and tensor-product elements, they have not been extended to cut meshes. Cut meshes provide a convenient and efficient means of mesh generation for domains with embedded boundaries but can be difficult to use due to their arbitrarily shaped cut elements. Using the skew-hybridized SBP formulation of Chan ["Skew-symmetric entropy stable...", JSC, 2019], we present a high-order accurate, entropy stable scheme for hyperbolic conservation laws on cut meshes. The formulation requires positive/non-negative weight quadrature rules on cut elements, which we construct via explicit parameterizations, subtriangulations, and Caratheodory pruning. We numerically verify the accuracy and stability of our method using the shallow water and compressible Euler equations and note promising results for the use of state redistribution with entropy stable methods.

math.NA↗

arXiv2024

A robust first order meshfree method for time-dependent nonlinear conservation laws

We introduce a robust first order accurate meshfree method to numerically solve time-dependent nonlinear conservation laws. The main contribution of this work is the meshfree construction of first order consistent summation by parts differentiations. We describe how to efficiently construct such operators on a point cloud. We then study the performance of such differentiations, and then combine these operators with a numerical flux-based formulation to approximate the solution of nonlinear conservation laws, with focus on the advection equation and the compressible Euler equations. We observe numerically that, while the resulting mesh-free differentiation operators are only $O(h^\frac{1}{2})$ accurate in the $L^2$ norm, they achieve $O(h)$ rates of convergence when applied to the numerical solution of PDEs.

math.NA↗

arXiv2024

An Energy Stable High-Order Cut Cell Discontinuous Galerkin Method with State Redistribution for Wave Propagation

Cut meshes are a type of mesh that is formed by allowing embedded boundaries to "cut" a simple underlying mesh resulting in a hybrid mesh of cut and standard elements. While cut meshes can allow complex boundaries to be represented well regardless of the mesh resolution, their arbitrarily shaped and sized cut elements can present issues such as the small cell problem, where small cut elements can result in a severely restricted CFL condition. State redistribution, a technique developed by Berger and Giuliani [1], can be used to address the small cell problem. In this work, we pair state redistribution with a high-order discontinuous Galerkin scheme that is $L_2$ energy stable for arbitrary quadrature. We prove that state redistribution can be added to a provably $L_2$ energy stable discontinuous Galerkin method on a cut mesh without damaging the scheme's $L_2$ stability. We numerically verify the high order accuracy and stability of our scheme on two-dimensional wave propagation problems.

math.NA↗

arXiv2023

Pseudodifferential Models for Ultrasound Waves with Fractional Attenuation

To strike a balance between modeling accuracy and computational efficiency for simulations of ultrasound waves in soft tissues, we derive a pseudodifferential factorization of the wave operator with fractional attenuation. This factorization allows us to approximately solve the Helmholtz equation via one-way (transmission) or two-way (transmission and reflection) sweeping schemes tailored to high-frequency wave fields. We provide explicitly the three highest order terms of the pseudodifferential expansion to incorporate the well-known square-root first order symbol for wave propagation, the zeroth order symbol for amplitude modulation due to changes in wave speed and damping, and the next symbol to model fractional attenuation. We also propose wide-angle Pade approximations for the pseudodifferential operators corresponding to these three highest order symbols. Our analysis provides insights regarding the role played by the frequency and the Pade approximations in the estimation of error bounds. We also provide a proof-of-concept numerical implementation of the proposed method and test the error estimates numerically.

math.NA↗

arXiv2023

High order entropy stable schemes for the quasi-one-dimensional shallow water and compressible Euler equations

High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features for nonlinear conservation laws. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of discretization parameters. This work extends high order entropy stable schemes to the quasi-1D shallow water equations and the quasi-1D compressible Euler equations, which model one-dimensional flows through channels or nozzles with varying width. We introduce new non-symmetric entropy conservative finite volume fluxes for both sets of quasi-1D equations, as well as a generalization of the entropy conservation condition to non-symmetric fluxes. When combined with an entropy stable interface flux, the resulting schemes are high order accurate, conservative, and semi-discretely entropy stable. For the quasi-1D shallow water equations, the resulting schemes are also well-balanced.

math.NA↗

arXiv2023

High order entropy stable discontinuous Galerkin spectral element methods through subcell limiting

Subcell limiting strategies for discontinuous Galerkin spectral element methods do not provably satisfy a semi-discrete cell entropy inequality. In this work, we introduce an extension to the subcell and monolithic convex limiting strategies that satisfies the semi-discrete cell entropy inequality by formulating the limiting factors as solutions to an optimization problem. The optimization problem is efficiently solved using a deterministic greedy algorithm. We also discuss the extension of the proposed subcell limiting strategy to preserve general convex constraints. Numerical experiments confirm that the proposed limiting strategy preserves high-order accuracy for smooth solutions and satisfies the cell entropy inequality.

math.NA↗

arXiv2022

High-Order Methods for Hypersonic Flows with Strong Shocks and Real Chemistry

We compare high-order methods including spectral difference (SD), flux reconstruction (FR), the entropy-stable discontinuous Galerkin spectral element method (ES-DGSEM), modal discontinuous Galerkin methods, and WENO to select the best candidate to simulate strong shock waves characteristic of hypersonic flows. We consider several benchmarks, including the Leblanc and modified shock-density wave interaction problems that require robust stabilization and positivity-preserving properties for a successful flow realization. We also perform simulations of the three-species Sod problem with simplified chemistry with the chemical reaction source terms introduced in the Euler equations. The ES-DGSEM scheme exhibits the highest stability, negligible numerical oscillations, and requires the least computational effort in resolving reactive flow regimes with strong shock waves. Therefore, we extend the ES-DGSEM to hypersonic Euler equations by deriving a new set of two-point entropy conservative fluxes for a five-species gas model. Stabilization for capturing strong shock waves occurs by blending high-order entropy conservative fluxes with low-order finite volume fluxes constructed using the HLLC Riemann solver. The hypersonic Euler solver is verified using the non-equilibrium chemistry Sod problem. To this end, we adopt the Mutation++ library to compute the reaction source terms, thermodynamic properties, and transport coefficients. We also investigate the effect of real chemistry versus ideal chemistry, and the results demonstrate that the ideal chemistry assumption fails at high temperatures, hence real chemistry must be employed for accurate predictions. Finally, we consider a viscous hypersonic flow problem to verify the transport coefficients and reaction source terms determined by the Mutation++ library.

physics.flu-dyn↗

arXiv2022

On the entropy projection and the robustness of high order entropy stable discontinuous Galerkin schemes for under-resolved flows

High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with under-resolved features. We demonstrate numerically that entropy stable DG methods which incorporate an "entropy projection" are less likely to require additional limiting to retain positivity for certain types of flows. We conclude by investigating potential explanations for this observed improvement in robustness.

math.NA↗

arXiv2022

Entropy-Stable Gauss Collocation Methods for Ideal Magneto-Hydrodynamics

In this paper, we present an entropy-stable Gauss collocation discontinuous Galerkin (DG) method on 3D curvilinear meshes for the GLM-MHD equations: the single-fluid magneto-hydrodynamics (MHD) equations with a generalized Lagrange multiplier (GLM) divergence cleaning mechanism. For the continuous entropy analysis to hold and to ensure Galilean invariance in the divergence cleaning technique, the GLM-MHD system requires the use of non-conservative terms. Traditionally, entropy-stable DG discretizations have used a collocated nodal variant of the DG method, also known as the discontinuous Galerkin spectral element method (DGSEM) on Legendre-Gauss-Lobatto (LGL) points. Recently, Chan et al. ("Efficient Entropy Stable Gauss Collocation Methods". SIAM -2019) presented an entropy-stable DGSEM scheme that uses Legendre-Gauss points (instead of LGL points) for conservation laws. Our main contribution is to extend the discretization technique of Chan et al. to the non-conservative GLM-MHD system. We provide a numerical verification of the entropy behavior and convergence properties of our novel scheme on 3D curvilinear meshes. Moreover, we test the robustness and accuracy of our scheme with a magneto-hydrodynamic Kelvin-Helmholtz instability problem. The numerical experiments suggest that the entropy-stable DGSEM on Gauss points for the GLM-MHD system is more accurate than the LGL counterpart.

math.NA↗