arXiv subjects

Shaoshuai Chu

Publications and source records attributed to Shaoshuai Chu.

At least 19 recordsLinked to original sources

arXiv2026

Numerical Realization of an Entropy-Based Selection Principle in the Example of Kelvin-Helmholtz Instability

In this paper, we extend numerical schemes based on parameterized Young measures and linear programming to two space dimensions and apply them to the Kelvin-Helmholtz instability governed by the two-dimensional Euler equations of gas dynamics. We construct first-, second-, third-, fifth-, seventh-, and ninth-order Young-measure schemes and compare them with corresponding standard local Lax-Friedrichs (LLF) flux-splitting schemes and LLF schemes equipped with an entropy-based local characteristic decomposition (LLF-ELCD). We examine instantaneous and time-averaged density profiles, cumulative averages across spatial reconstruction orders, regional empirical density distributions, averaged density marginals, and several selection criteria for dissipative weak solutions. The numerical results reveal scheme-dependent flow patterns, particularly in the small-scale structures generated during the roll-up of the shear layers. At every reconstruction order considered, the Young-measure schemes yield the largest time-averaged physical entropy. By contrast, the LLF-ELCD schemes do not systematically yield larger entropy than the standard LLF schemes and therefore do not constitute a consistent maximum-entropy selection mechanism. The averaged density marginals of the Young measures are concentrated on several neighboring density states, and the later cumulative averages are closer for all Young-measure schemes. These results demonstrate the feasibility of the two-dimensional Young-measure formulation and show that the objective function in the linear-programming problem can act as an effective selection mechanism. The observed entropy preference is consistent with the local optimization of the expected physical entropy. It cannot be reproduced by merely incorporating entropy into a conventional numerical construction.

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arXiv2026

Entropy-Based Local Characteristic Decomposition

Local characteristic decomposition (LCD) is widely used in high-order numerical methods for hyperbolic systems of conservation laws to reduce spurious oscillations appearing in the computed solutions. The LCD implementation requires a representative average interface state, typically obtained using arithmetic or Roe-type averages of the nearly grid values. We propose an entropy-based LCD (ELCD), in which the states in the two cells adjacent to an interface are considered together with several nearby states in phase space. Among these candidates, we select the state that locally minimizes the entropy and use it as an average interface state for linearizing the flux Jacobian. We incorporate the ELCD into several second-order finite-volume and fifth-order finite-difference schemes. Numerical experiments for the two-dimensional Euler equations of gas dynamics show that the schemes, which utilize the ELCD procedure generally resolve complex wave structures more sharply than their counterparts, which use the arithmetic averages.

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arXiv2026

A Comparative Study of Low-Dissipation Numerical Schemes for Hyperbolic Conservation Laws

This work provides a comparative assessment of several low-dissipation numerical schemes for hyperbolic conservation laws, highlighting their performance relative to the classical Harten-Lax-van Leer (HLL) schemes. The schemes under consideration include the classical Harten-Lax-van Leer-Contact (HLLC), the recently proposed TV flux splitting, the low-dissipation Central-Upwind (LDCU), and the local characteristic decomposition-based Central-Upwind (LCDCU) schemes. These methods are extended to higher orders of accuracy, up to the fifth order, within both finite-volume and finite-difference frameworks. A series of numerical experiments for the one- and two-dimensional Euler equations of gas dynamics are performed to evaluate the accuracy, robustness, and computational efficiency of the studied schemes. The comparison highlights the trade-offs between resolution of contact and shear waves, robustness in the presence of shocks, and computational cost. The investigated low-dissipation schemes show comparable levels of numerical dissipation, with only subtle differences appearing in selected benchmark problems. The results provide practical guidance for selecting efficient low-dissipation solvers for the simulation of complex compressible flows.

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arXiv2026

Novel Adaptive Methods for Hyperbolic Conservation Laws Based on New Quasi-Linear Seventh- and Ninth-Order Schemes

We develop new adaptive numerical schemes for one- and two-dimensional hyperbolic systems of conservation laws. The methodology relies on the use of a smoothness indicator to automatically partition the computational domain into smooth and nonsmooth (``rough``) regions. We then follow the scheme adaption strategy recently introduced in [S. Chu, P. Feng, V. A. Kolotilov, A. Kurganov, and V. V. Ostapenko, Commun. Comput. Phys., accepted], but instead of the quasi-linear (QL) fifth-order finite-difference scheme used there, we employ the new QL seventh- and ninth-order schemes in the smooth regions. A series of numerical experiments for the Euler equations of gas dynamics demonstrates that the new adaptive schemes contain a smaller amount of numerical dissipation and achieve higher resolution compared with their counterpart that uses the QL fifth-order scheme in the smooth areas.

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arXiv2026

Local Characteristic Decomposition of Equilibrium Variables for Hyperbolic Systems of Balance Laws

This paper is concerned with high-order numerical methods for hyperbolic systems of balance laws. Such methods are typically based on high-order piecewise polynomial reconstructions (interpolations) of the computed discrete quantities. However, such reconstructions (interpolations) may be oscillatory unless the reconstruction (interpolation) procedure is applied to the local characteristic variables via the local characteristic decomposition (LCD). Another challenge in designing accurate and stable high-order schemes is related to enforcing a delicate balance between the fluxes, sources, and nonconservative product terms: a good scheme should be well-balanced (WB) in the sense that it should be capable of exactly preserving certain (physically relevant) steady states. One of the ways to ensure that the reconstruction (interpolation) preserves these steady states is to apply the reconstruction (interpolation) to the equilibrium variables, which are supposed to be constant at the steady states. To achieve this goal and to keep the reconstruction (interpolation) non-oscillatory, we introduce a new LCD of equilibrium variables. We apply the developed technique to the fifth-order Ai-WENO-Z interpolation implemented within the WB A-WENO framework recently introduced in [S. Chu, A. Kurganov, and R. Xin, Beijing J. of Pure and Appl. Math., 2 (2025), pp. 87--113], and illustrate its performance on a variety of numerical examples.

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arXiv2026

Adaptive Artificial Anti-Diffusion Methods for Hyperbolic Systems of Conservation Laws

We introduce new adaptive artificial anti-diffusion (AAAD) methods for one- and two-dimensional hyperbolic systems of conservation laws. The key idea is to reduce the amount of numerical dissipation present in a given numerical method by adding an anti-diffusion (AD) term acting in linearly degenerate fields only. This way, the resolution of contact waves can be improved without risking oscillations, which may be caused if the AD acts in nonlinear fields as well. The AD coefficients are selected adaptively: they are supposed to be proportional to the mesh size near the contact waves to enhance the resolution and to be very small in the smooth parts of the computed solution to ensure a sufficiently high (formal) order of accuracy there. The proposed AAAD methods are realized using either the second-order central-upwind numerical fluxes or their fifth-order extensions implemented within the alternative weighted essentially non-oscillatory (A-WENO) framework. We test the proposed schemes on a series of benchmarks for the one- and two-dimensional Euler equations of gas dynamics and the obtained results demonstrate the robustness and high resolution of the new AAAD methods.

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arXiv2026

New Scheme Adaption Strategy for Hyperbolic Conservation Laws

We introduce a new scheme adaption strategy for one- and two-dimensional hyperbolic systems of conservation laws. The proposed approach builds upon the adaptive framework introduced in [S. Chu, A. Kurganov, and I. Menshov, Appl. Numer. Math., 209 (2025), pp.155--170], where we first employed the smoothness indicator from [R. Lohner, Comput. Methods. Appl. Mech. Eng., 61 (1987), pp.323--338] to automatically detect ``rough'' and smooth parts of the computed solution, and then used different limiters in the detected regions. This adaptive strategy was based on a threshold needed to sharply separate ``rough'' and smooth regions. In this paper, we propose a different adaption strategy. We use SBM-type limiters and vary one of the limiting parameters continuously to allow a smooth transition between the ``rough'' and smooth areas. This way, compressive and overcompressive limiters are activated in the shock and contact wave vicinities only, while we gradually switch to dissipative limiters in the smooth regions. A series of one- and two-dimensional numerical tests for the Euler equations of gas dynamics demonstrates that the new scheme adaption strategy leads to a higher resolution and reduced numerical dissipation.

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arXiv2026

A Bayesian Approach to Feedback Control for Hyperbolic Balance Laws

We propose a Bayesian framework for feedback boundary control of hyperbolic balance laws. The method propagates a probability distribution over feedback parameters using Lyapunov decay estimates as a likelihood. For linear models, it recovers available analytical stability results and extends to nonlinear regimes where theory is limited. Using first-order local Lax-Friedrichs (LLF) discretizations, we validate the approach on the decoupled wave system and the linearized Saint-Venant equations, reproducing known stability intervals and mixed boundary couplings. We then treat nonlinear and stochastic problems, including the nonlinear Saint-Venant system, one- and two-dimensional Burgers equations, Burgers equation with random initial data, and nonconservative perturbations with source terms, and show that the inferred stability domains are robust with respect to the indicator and the prior. Finally, we demonstrate transfer to a second-order semi-discrete LLF scheme and to a two-parameter feedback model for laser powder bed fusion with power regulation.

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arXiv2026

Numerical Study of Dissipative Weak Solutions for the Euler Equations of Gas Dynamics

We study dissipative weak (DW) solutions of the Euler equations of gas dynamics using the first-, second-, third-, fifth-, seventh-, and ninth-order local characteristic decomposition-based central-upwind (LCDCU), low-dissipation central-upwind (LDCU), and viscous finite volume (VFV) methods, whose higher-order extensions are obtained via the framework of the alternative weighted essentially non-oscillatory (A-WENO) schemes. These methods are applied to several benchmark problems, including several two-dimensional Riemann problems and a Kelvin-Helmholtz instability test. The numerical results demonstrate that for methods converging only weakly in space and time, the limiting solutions are generalized DW solutions, approximated in the sense of ${\cal K}$-convergence and dependent on the numerical scheme. For all of the studied methods, we compute the associated Young measures and compare the DW solutions using entropy production and energy defect criteria.

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arXiv2026

High-Order Schemes for Hyperbolic Conservation Laws Using Young Measures

We develop high-order numerical schemes to solve random hyperbolic conservation laws using linear programming. The proposed schemes are high-order extensions of the existing first-order scheme introduced in [{\sc S. Chu, M. Herty, M. Lukáčová-Medvi{\softd}ová, and Y. Zhou}, SIAM J. Sci. Comput., 48 (2026)], where a novel structure-preserving numerical method using a concept of generalized, measure-valued solutions to solve random hyperbolic systems of conservation laws is proposed, yielding a linear partial differential equation concerning the Young measure and allowing the computation of approximations based on linear programming problems. The second-order extension is obtained using piecewise linear reconstructions of the one-sided point values of the unknowns. The fifth-order scheme is developed using the finite-difference alternative weighted essentially non-oscillatory (A-WENO) framework. These extensions significantly improve the resolution of discontinuities, as demonstrated by a series of numerical experiments on both random (Burgers equation, isentropic Euler equations) and deterministic (discontinuous flux, pressureless gas dynamics, Burgers equation with non-atomic support) hyperbolic conservation laws.

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arXiv2026

Novel Adaptive Schemes for Hyperbolic Conservation Laws

We introduce new adaptive schemes for the one- and two-dimensional hyperbolic systems of conservation laws. Our schemes are based on an adaption strategy recently introduced in [{\sc S. Chu, A. Kurganov, and I. Menshov}, Appl. Numer. Math., 209 (2025)]. As there, we use a smoothness indicator (SI) to automatically detect ``rough'' parts of the solution and employ in those areas the second-order finite-volume low-dissipation central-upwind scheme with an overcompressive limiter, which helps to sharply resolve nonlinear shock waves and linearly degenerate contact discontinuities. In smooth parts, we replace the limited second-order scheme with a quasi-linear fifth-order (in space and third-order in time) finite-difference scheme, recently proposed in [{\sc V. A. Kolotilov, V. V. Ostapenko, and N. A. Khandeeva}, Comput. Math. Math. Phys., 65 (2025)]. However, direct application of this scheme may generate spurious oscillations near ``rough'' parts, while excessive use of the overcompressive limiter may cause staircase-like nonphysical structures in smooth areas. To address these issues, we employ the same SI to distinguish contact discontinuities, treated with the overcompressive limiter, from other ``rough'' regions, where we switch to the dissipative Minmod2 limiter. Advantage of the resulting adaptive schemes are clearly demonstrated on a number of challenging numerical examples.

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arXiv2025

A Locally Divergence-Free Local Characteristic Decomposition Based Path-Conservative Central-Upwind Scheme for Ideal Magnetohydrodynamics

We introduce a locally divergence-free local characteristic decomposition based path-conservative central-upwind (LCD-PCCU) scheme for ideal magnetohydrodynamics (MHD) equations. The proposed method is a low-dissipation extension of the recently proposed locally divergence-free PCCU scheme. To reduce the numerical dissipation, we incorporate the LCD into the PCCU framework. The resulting LCD-PCCU method enhances the resolution of numerical solutions as demonstrated through a series of benchmark tests.

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arXiv2025

New Fully Discrete Active Flux Methods with Truly Multi-Dimensional Evolution Operators and WENO Reconstruction

We propose new fully discrete third-order accurate Active Flux and WENO methods based on truly multidimensional evolution operators for the two-dimensional acoustic equations. Building on the method of bicharacteristics, several approximate evolution operators are derived that yield an improved stability of the resulting schemes. A linear stability analysis is applied to determine the maximal CFL number. The schemes are tested extensively on both continuous and discontinuous problems, confirming their robustness and accurate approximation even on coarse grids.

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arXiv2025

Solving Random Hyperbolic Conservation Laws Using Linear Programming

A novel structure-preserving numerical method to solve random hyperbolic systems of conservation laws is presented. The method uses a concept of generalized, measure-valued solutions to random conservation laws. This yields a linear partial differential equation with respect to the Young measure and allows to compute the approximation based on linear programming problems. We analyze structure-preserving properties of the derived numerical method and discuss its advantages and disadvantages. We numerically demonstrate the approach on the one-dimensional Burgers and isentropic Euler equations and compare with stochastic collocation. In addition, we introduce a discontinuous-flux test in which different entropies used in the linear-program objective select different weak entropy solutions, and we report the corresponding changes in the moments and supports of the Young measure.

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arXiv2025

Numerical Analysis of Stabilization for Random Hyperbolic Systems of Conservation Laws

This paper extends the deterministic Lyapunov-based stabilization framework to random hyperbolic systems of conservation laws, where uncertainties arise in boundary controls and initial data. Building on the finite volume discretization method from [{\sc M. Banda and M. Herty}, Math. Control Relat. Fields., 3 (2013), pp. 121--142], we introduce a stochastic discrete Lyapunov function to prove the exponential decay of numerical solutions for systems with random perturbations. For linear systems, we derive explicit decay rates, which depend on boundary control parameters, grid resolutions, and the statistical properties of the random inputs. Theoretical decay rates are verified through numerical examples, including boundary stabilization of the linear wave equations and linearized shallow-water flows with random perturbations. We also present the decay rates for a nonlinear example and for the linearized Saint-Venant system with source terms.

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arXiv2025

High-Order Flux Splitting Schemes for the Euler Equations of Gas Dynamics

We develop high-order flux splitting schemes for the one- and two-dimensional Euler equations of gas dynamics. The proposed schemes are high-order extensions of the existing first-order flux splitting schemes introduced in [ E. F. Toro, M. E. Vázquez-Cendón, Comput. \& Fluids, 70 (2012), pp. 1--12], where the Euler equations of gas dynamics are split into two subsystems: the advection and pressure systems. In this paper, we formulate the TV splitting within the semi-discrete framework to extend it to higher orders of accuracy for the first time. The second-order extension is obtained by using piecewise linear interpolant to reconstruct the one-sided point values of the unknowns. The third- and fifth-order schemes are developed using the finite-difference alternative weighted essentially non-oscillatory (A-WENO) framework, which is particularly effective in handling multidimensional problems and provides a more straightforward approach to constructing higher-order WENO schemes. These extensions significantly improve the resolution of discontinuities and the accuracy of numerical solutions, as demonstrated by a series of numerical experiments of both the one- and two-dimensional Euler equations of gas dynamics.

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arXiv2025

Central-Upwind Scheme for the Phase-Transition Traffic Flow Model

Phase-transition models are an important family of non-equilibrium continuum traffic flow models, offering properties like replicating complex traffic phenomena, maintaining anisotropy, and promising potentials for accommodating automated vehicles. However, their complex mathematical characteristics such as discontinuous solution domains, pose numerical challenges and limit their exploration in traffic flow theory. This paper focuses on developing a robust and accurate numerical method for phase-transition traffic flow models: We propose a second-order semi-discrete central-upwind scheme specifically designed for discontinuous phase-transition models. This novel scheme incorporates the projection onto appropriate flow domains, ensuring enhanced handling of discontinuities and maintaining physical consistency and accuracy. We demonstrate the efficacy of the proposed scheme through extensive and challenging numerical tests, showcasing their potential to facilitate further research and application in phase-transition traffic flow modeling. The ability of phase-transition models to embed the ``time-gap'' -- a crucial element in automated traffic control -- as a conserved variable aligns seamlessly with the control logic of automated vehicles, presenting significant potential for future applications, and the proposed numerical scheme now substantially facilitates exploring such potentials.

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arXiv2025

New More Efficient A-WENO Schemes

We develop new more efficient A-WENO schemes for both hyperbolic systems of conservation laws and nonconservative hyperbolic systems. The new schemes are a very simple modification of the existing A-WENO schemes: They are obtained by a more efficient evaluation of the high-order correction terms. We conduct several numerical experiments to demonstrate the performance of the introduced schemes.

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