arXiv subjects

Kailiang Wu

Publications and source records attributed to Kailiang Wu.

At least 37 records · Page 2Linked to original sources

arXiv2025

On Local Minimum Entropy Principle of High-Order Schemes for Relativistic Euler Equations

This paper establishes the minimum entropy principle (MEP) for the relativistic Euler equations with a broad class of equations of state (EOSs) and addresses the challenge of preserving the local version of the discovered MEP in high-order numerical schemes. At the continuous level, we find out a family of entropy pairs for the relativistic Euler equations and provide rigorous analysis to prove the strict convexity of entropy under a necessary and sufficient condition. At the numerical level, we develop a rigorous framework for designing provably entropy-preserving high-order schemes that ensure both physical admissibility and the discovered MEP. The relativistic effects, coupled with the abstract and general EOS formulation, introduce significant challenges not encountered in the nonrelativistic case or with the ideal EOS. In particular, entropy is a highly nonlinear and implicit function of the conservative variables, making it particularly difficult to enforce entropy preservation. To address these challenges, we establish a series of auxiliary theories via highly technical inequalities. Another key innovation is the use of geometric quasi-linearization (GQL), which reformulates the nonlinear constraints into equivalent linear ones by introducing additional free parameters. These advancements form the foundation of our entropy-preserving analysis. We propose novel, robust, locally entropy-preserving high-order frameworks. A central challenge is accurately estimating the local minimum of entropy, particularly in the presence of shock waves at unknown locations. To address this, we introduce two new approaches for estimating local lower bounds of specific entropy, which prove effective for both smooth and discontinuous problems. Numerical experiments demonstrate that our entropy-preserving methods maintain high-order accuracy while effectively suppressing spurious oscillations.

math.NA↗

arXiv2025

Oscillation-eliminating central DG schemes for hyperbolic conservation laws

This paper proposes and analyzes a class of essentially non-oscillatory central discontinuous Galerkin (CDG) methods for general hyperbolic conservation laws. First, we introduce a novel compact, non-oscillatory stabilization mechanism that effectively suppresses spurious oscillations while preserving the high-order accuracy of CDG methods. Unlike existing limiter-based approaches that rely on large stencils or problem-specific parameters for oscillation control, our dual damping mechanism is inspired by CDG-based numerical dissipation and leverages overlapping solutions within the CDG framework, significantly enhancing stability while maintaining compactness. Our approach is free of problem-dependent parameters and complex characteristic decomposition, making it efficient and robust. Second, we provide a rigorous stability and optimal error analysis for fully discrete Runge-Kutta (RK) CDG schemes, addressing a gap in the theoretical understanding of these methods. Specifically, we establish the approximate skew-symmetry and weak boundedness of the CDG discretization. These results enable us to rigorously analyze the fully discrete error estimates for our oscillation-eliminating CDG (OECDG) method, a challenging task due to its nonlinear nature, even for linear advection equations. Building on this framework, we reformulate nonlinear oscillation-eliminating CDG schemes as linear RK CDG schemes with a nonlinear source term, extending error estimates beyond the linear case to schemes with nonlinear oscillation control. While existing error analyses for DG or CDG schemes have largely been restricted to linear cases without nonlinear oscillation-control techniques, our analysis represents an important theoretical advancement. Experiments validate the theoretical findings and demonstrate the effectiveness of the OECDG method.

math.NA↗

arXiv2024

A Novel and Simple Invariant-Domain-Preserving Framework for PAMPA Scheme: 1D Case

The PAMPA (Point-Average-Moment PolynomiAl-interpreted) method, proposed in [R. Abgrall, Commun. Appl. Math. Comput., 5: 370-402, 2023], combines conservative and non-conservative formulations of hyperbolic conservation laws to evolve cell averages and point values. Solutions to hyperbolic conservation laws typically have an invariant domain, and ensuring numerical solutions stay within this domain is essential yet nontrivial. This paper presents a novel framework for designing efficient Invariant-Domain-Preserving (IDP) PAMPA schemes. We first analyze the IDP property for updated cell averages in the original PAMPA scheme, revealing the role of cell average decomposition and midpoint values in maintaining the invariant domain. This analysis highlights the difficulty of relying on continuous fluxes alone to preserve the invariant domain. Building on these insights, we introduce a simple IDP limiter for cell midpoint values, and propose a provably IDP PAMPA scheme that guarantees the preservation of the invariant domain for updated cell averages without requiring post-processing limiters. This approach contrasts with existing bound-preserving PAMPA schemes, which often require additional convex limiting to blend high-order and low-order solutions. Most notably, inspired by the Softplus and Clipped ReLU functions from machine learning, we propose an automatic IDP reformulation of the governing equations, resulting in an unconditionally limiter-free IDP scheme for evolving point values. We also introduce techniques to suppress spurious oscillations, enabling the scheme to capture strong shocks effectively. Numerical experiments on 1D problems, including the linear convection equation, Burgers equation, the compressible Euler equations, and MHD equations, demonstrate the accuracy and robustness of the proposed IDP PAMPA scheme.

math.NA↗

arXiv2024

Robust Discontinuous Galerkin Methods Maintaining Physical Constraints for General Relativistic Hydrodynamics

Simulating general relativistic hydrodynamics (GRHD) presents challenges such as handling curved spacetime, achieving high-order shock-capturing accuracy, and preserving key physical constraints (positive density, pressure, and subluminal velocity) under nonlinear coupling. This paper introduces high-order, physical-constraint-preserving, oscillation-eliminating discontinuous Galerkin (PCP-OEDG) schemes with Harten-Lax-van Leer flux for GRHD. To suppress spurious oscillations near discontinuities, we incorporate a computationally efficient oscillation-eliminating (OE) procedure based on a linear damping equation, maintaining accuracy and avoiding complex characteristic decomposition. To enhance stability and robustness, we construct PCP schemes using the W-form of GRHD equations with Cholesky decomposition of the spatial metric, addressing the non-equivalence of admissible state sets in curved spacetime. We rigorously prove the PCP property of cell averages via technical estimates and the Geometric Quasi-Linearization (GQL) approach, which transforms nonlinear constraints into linear forms. Additionally, we present provably convergent PCP iterative algorithms for robust recovery of primitive variables, ensuring physical constraints are satisfied throughout. The PCP-OEDG method is validated through extensive tests, demonstrating its robustness, accuracy, and capability to handle extreme GRHD scenarios involving strong shocks, high Lorentz factors, and intense gravitational fields.

math.NA↗

arXiv2024

Provably Positivity-Preserving Constrained Transport (PPCT) Second-Order Scheme for Ideal Magnetohydrodynamics

This paper proposes and analyzes a robust and efficient second-order positivity-preserving constrained transport (PPCT) scheme for ideal magnetohydrodynamics (MHD) on non-staggered Cartesian meshes. The PPCT scheme ensures two critical physical constraints: a globally discrete divergence-free (DDF) condition on the magnetic field and the positivity of density and pressure. The method is inspired by a novel splitting technique from [T.A. Dao, M. Nazarov and I. Tomas, J. Comput. Phys., 508:113009, 2024], which divides the MHD system into an Euler subsystem with steady magnetic fields and a magnetic subsystem with steady density and internal energy. To achieve these structure-preserving properties, the PPCT scheme combines a positivity-preserving (PP) finite volume method for the Euler subsystem with a finite difference constrained transport (CT) method for the magnetic subsystem via Strang splitting. The finite volume method employs a new PP limiter that retains second-order accuracy and enforces the positivity of density and pressure, with rigorous proof provided using the geometric quasilinearization (GQL) approach [K. Wu and C.-W. Shu, SIAM Review, 65:1031-1073, 2023]. For the magnetic subsystem, we develop an implicit finite difference CT method that conserves energy and maintains a globally DDF constraint. This nonlinear system is efficiently solved to machine precision using an iterative algorithm. Since the CT method is unconditionally energy-stable and conserves steady density and internal energy, the PPCT scheme requires only a mild CFL condition for the finite volume method to ensure stability and the PP property. While the focus is on 2D cases for clarity, the extension to 3D is discussed. Several challenging numerical experiments, including highly magnetized MHD jets with high Mach numbers, validate the PPCT scheme's accuracy, robustness, and high resolution.

math.NA↗

arXiv2024

Bound-preserving OEDG schemes for Aw-Rascle-Zhang traffic models on networks

Physical solutions to the widely used Aw-Rascle-Zhang (ARZ) traffic model and the adapted pressure (AP) ARZ model should satisfy the positivity of density, the minimum and maximum principles with respect to the velocity $v$ and other Riemann invariants. Many numerical schemes suffer from instabilities caused by violating these bounds, and the only existing bound-preserving (BP) numerical scheme (for ARZ model) is random, only first-order accurate, and not strictly conservative. This paper introduces arbitrarily high-order provably BP DG schemes for these two models, preserving all the aforementioned bounds except the maximum principle of $v$, which has been rigorously proven to conflict with the consistency and conservation of numerical schemes. Although the maximum principle of $v$ is not directly enforced, we find that the strictly preserved maximum principle of another Riemann invariant $w$ actually enforces an alternative upper bound on $v$. At the core of this work, analyzing and rigorously proving the BP property is a particularly nontrivial task: the Lax-Friedrichs (LF) splitting property, usually expected for hyperbolic conservation laws and employed to construct BP schemes, does not hold for these two models. To overcome this challenge, we formulate a generalized version of the LF splitting property, and prove it via the geometric quasilinearization (GQL) approach [Kailiang Wu and Chi-Wang Shu, SIAM Review, 65: 1031-1073, 2023]. To suppress spurious oscillations in the DG solutions, we employ the oscillation-eliminating (OE) technique, recently proposed in [Manting Peng, Zheng Sun, and Kailiang Wu, Mathematics of Computation, in press], which is based on the solution operator of a novel damping equation. Several numerical examples are included to demonstrate the effectiveness, accuracy, and BP properties of our schemes, with applications to traffic simulations on road networks.

math.NA↗

arXiv2024

Spectral Volume from a DG perspective: Oscillation Elimination, Stability, and Optimal Error Estimates

The discontinuous Galerkin (DG) method and the spectral volume (SV) method are two widely-used numerical methodologies for solving hyperbolic conservation laws. In this paper, we demonstrate that under specific subdivision assumptions, the SV method can be represented in a DG form with a different inner product. Building on this insight, we extend the oscillation-eliminating (OE) technique, recently proposed in [M. Peng, Z. Sun, and K. Wu, {\it Mathematics of Computation}, https://doi.org/10.1090/mcom/3998], to develop a new fully-discrete OESV method. The OE technique is non-intrusive, efficient, and straightforward to implement, acting as a simple post-processing filter to effectively suppress spurious oscillations. From a DG perspective, we present a comprehensive framework to theoretically analyze the stability and accuracy of both general Runge-Kutta SV (RKSV) schemes and the novel OESV method. For the linear advection equation, we conduct an energy analysis of the fully-discrete RKSV method, identifying an upwind condition crucial for stability. Furthermore, we establish optimal error estimates for the OESV schemes, overcoming nonlinear challenges through error decomposition and treating the OE procedure as additional source terms in the RKSV schemes. Extensive numerical experiments validate our theoretical findings and demonstrate the effectiveness and robustness of the proposed OESV method. This work enhances the theoretical understanding and practical application of SV schemes for hyperbolic conservation laws, making the OESV method a promising approach for high-resolution simulations.

math.NA↗

arXiv2024

High-order Accurate Entropy Stable Schemes for Relativistic Hydrodynamics with General Synge-type Equation of State

All the existing entropy stable (ES) schemes for relativistic hydrodynamics (RHD) in the literature were restricted to the ideal equation of state (EOS), which however is often a poor approximation for most relativistic flows due to its inconsistency with the relativistic kinetic theory. This paper develops high-order ES finite difference schemes for RHD with general Synge-type EOS, which encompasses a range of special EOSs. We first establish an entropy pair for the RHD equations with general Synge-type EOS in any space dimensions. We rigorously prove that the found entropy function is strictly convex and derive the associated entropy variables, laying the foundation for designing entropy conservative (EC) and ES schemes. Due to relativistic effects, one cannot explicitly express primitive variables, fluxes, and entropy variables in terms of conservative variables. Consequently, this highly complicates the analysis of the entropy structure of the RHD equations, the investigation of entropy convexity, and the construction of EC numerical fluxes. By using a suitable set of parameter variables, we construct novel two-point EC fluxes in a unified form for general Synge-type EOS. We obtain high-order EC schemes through linear combinations of the two-point EC fluxes. Arbitrarily high-order accurate ES schemes are achieved by incorporating dissipation terms into the EC schemes, based on (weighted) essentially non-oscillatory reconstructions. Additionally, we derive the general dissipation matrix for general Synge-type EOS based on the scaled eigenvectors of the RHD system. We also define a suitable average of the dissipation matrix at the cell interfaces to ensure that the resulting ES schemes can resolve stationary contact discontinuities accurately. Several numerical examples are provided to validate the accuracy and effectiveness of our schemes for RHD with four special EOSs.

math.NA↗

arXiv2024

High-order accurate structure-preserving finite volume schemes on adaptive moving meshes for shallow water equations: Well-balancedness and positivity

This paper develops high-order accurate, well-balanced (WB), and positivity-preserving (PP) finite volume schemes for shallow water equations on adaptive moving structured meshes. The mesh movement poses new challenges in maintaining the WB property, which not only depends on the balance between flux gradients and source terms but is also affected by the mesh movement. To address these complexities, the WB property in curvilinear coordinates is decomposed into flux source balance and mesh movement balance. The flux source balance is achieved by suitable decomposition of the source terms, the numerical fluxes based on hydrostatic reconstruction, and appropriate discretization of the geometric conservation laws (GCLs). Concurrently, the mesh movement balance is maintained by integrating additional schemes to update the bottom topography during mesh adjustments. The proposed schemes are rigorously proven to maintain the WB property by using the discrete GCLs and these two balances. We provide rigorous analyses of the PP property under a sufficient condition enforced by a PP limiter. Due to the involvement of mesh metrics and movement, the analyses are nontrivial, while some standard techniques, such as splitting high-order schemes into convex combinations of formally first-order PP schemes, are not directly applicable. Various numerical examples validate the high-order accuracy, high efficiency, WB, and PP properties of the proposed schemes.

math.NA↗

arXiv2024

Robust DG Schemes on Unstructured Triangular Meshes: Oscillation Elimination and Bound Preservation via Optimal Convex Decomposition

Discontinuous Galerkin (DG) schemes on unstructured meshes offer the advantages of compactness and the ability to handle complex computational domains. However, their robustness and reliability in solving hyperbolic conservation laws depend on two critical abilities: suppressing spurious oscillations and preserving intrinsic bounds or constraints. This paper introduces two significant advancements in enhancing the robustness and efficiency of DG methods on unstructured meshes for general hyperbolic conservation laws, while maintaining their accuracy and compactness. First, we investigate the oscillation-eliminating (OE) DG methods on unstructured meshes. These methods not only maintain key features such as conservation, scale invariance, and evolution invariance but also achieve rotation invariance through a novel rotation-invariant OE (RIOE) procedure. Second, we propose, for the first time, the optimal convex decomposition for designing efficient bound-preserving (BP) DG schemes on unstructured meshes. Finding the optimal convex decomposition that maximizes the BP CFL number is an important yet challenging problem.While this challenge was addressed for rectangular meshes, it remains an open problem for triangular meshes. This paper successfully constructs the optimal convex decomposition for the widely used $P^1$ and $P^2$ spaces on triangular cells, significantly improving the efficiency of BP DG methods.The maximum BP CFL numbers are increased by 100%--200% for $P^1$ and 280.38%--350% for $P^2$, compared to classic decomposition. Furthermore, our RIOE procedure and optimal decomposition technique can be integrated into existing DG codes with little and localized modifications. These techniques require only edge-neighboring cell information, thereby retaining the compactness and high parallel efficiency of original DG methods.

math.NA↗

arXiv2024

High-Order Oscillation-Eliminating Hermite WENO Method for Hyperbolic Conservation Laws

This paper proposes high-order accurate, oscillation-eliminating Hermite weighted essentially non-oscillatory (OE-HWENO) finite volume schemes for hyperbolic conservation laws. The OE-HWENO schemes apply an OE procedure after each Runge--Kutta stage, dampening the first-order moments of the HWENO solution to suppress spurious oscillations without any problem-dependent parameters. This OE procedure acts as a filter, derived from the solution operator of a novel damping equation, solved exactly without discretization. As a result, the OE-HWENO method remains stable with a normal CFL number, even for strong shocks producing highly stiff damping terms. To ensure the method's non-oscillatory property across varying scales and wave speeds, we design a scale- and evolution-invariant damping equation and propose a dimensionless transformation for HWENO reconstruction. The OE-HWENO method offers several advantages over existing HWENO methods: the OE procedure is efficient and easy to implement, requiring only simple multiplication of first-order moments; it preserves high-order accuracy, local compactness, and spectral properties. The non-intrusive OE procedure can be integrated seamlessly into existing HWENO codes. Finally, we analyze the bound-preserving (BP) property using optimal cell average decomposition, relaxing the BP time step-size constraint and reducing decomposition points, improving efficiency. Extensive benchmarks validate the method's accuracy, efficiency, resolution, and robustness.

math.NA↗

arXiv2024

Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning

Operator learning for Partial Differential Equations (PDEs) is rapidly emerging as a promising approach for surrogate modeling of intricate systems. Transformers with the self-attention mechanism$\unicode{x2013}$a powerful tool originally designed for natural language processing$\unicode{x2013}$have recently been adapted for operator learning. However, they confront challenges, including high computational demands and limited interpretability. This raises a critical question: Is there a more efficient attention mechanism for Transformer-based operator learning? This paper proposes the Position-induced Transformer (PiT), built on an innovative position-attention mechanism, which demonstrates significant advantages over the classical self-attention in operator learning. Position-attention draws inspiration from numerical methods for PDEs. Different from self-attention, position-attention is induced by only the spatial interrelations of sampling positions for input functions of the operators, and does not rely on the input function values themselves, thereby greatly boosting efficiency. PiT exhibits superior performance over current state-of-the-art neural operators in a variety of complex operator learning tasks across diverse PDE benchmarks. Additionally, PiT possesses an enhanced discretization convergence feature, compared to the widely-used Fourier neural operator.

cs.LG↗

arXiv2024

Structure-Preserving Oscillation-Eliminating Discontinuous Galerkin Schemes for Ideal MHD Equations: Locally Divergence-Free and Positivity-Preserving

Numerically simulating magnetohydrodynamics (MHD) poses notable challenges, including the suppression of spurious oscillations near discontinuities (e.g., shocks) and preservation of essential physical structures (e.g., the divergence-free constraint of magnetic field and the positivity of density and pressure). This paper develops structure-preserving oscillation-eliminating discontinuous Galerkin (OEDG) schemes for ideal MHD. The schemes leverage a locally divergence-free (LDF) oscillation-eliminating (OE) procedure to suppress spurious oscillations while retaining the LDF property of magnetic field and many desirable attributes of original DG schemes, such as conservation, local compactness, and optimal convergence rates. The OE procedure is based on the solution operator of a novel damping equation, a linear system of ordinary differential equations that are exactly solvable without any discretization. The OE procedure is performed after each Runge-Kutta stage and does not impact DG spatial discretization, facilitating its easy integration into existing DG codes as an independent module. Moreover, this paper presents a rigorous positivity-preserving (PP) analysis of the LDF OEDG schemes on Cartesian meshes, utilizing the optimal convex decomposition technique and the geometric quasi-linearization (GQL) approach. Efficient PP LDF OEDG schemes are derived by incorporating appropriate discretization of Godunov-Powell source terms into only the discrete equations of cell averages, under a condition achievable through a simple PP limiter. Several one- and two-dimensional MHD tests verify the accuracy, effectiveness, and robustness of the proposed structure-preserving OEDG schemes.

math.NA↗

arXiv2024

Provably Convergent and Robust Newton-Raphson Method: A New Dawn in Primitive Variable Recovery for Relativistic MHD

A long-standing and formidable challenge faced by all conservative schemes for relativistic magnetohydrodynamics (RMHD) is the recovery of primitive variables from conservative ones. This process involves solving highly nonlinear equations subject to physical constraints. An ideal solver should be "robust, accurate, and fast -- it is at the heart of all conservative RMHD schemes," as emphasized in [S.C. Noble et al., ApJ, 641:626-637, 2006]. Despite over three decades of research, seeking efficient solvers that can provably guarantee stability and convergence remains an open problem. This paper presents the first theoretical analysis for designing a robust, physical-constraint-preserving (PCP), and provably (quadratically) convergent Newton-Raphson (NR) method for primitive variable recovery in RMHD. Our key innovation is a unified approach for the initial guess, devised based on sophisticated analysis. It ensures that the NR iteration consistently converges and adheres to physical constraints. Given the extreme nonlinearity and complexity of the iterative function, the theoretical analysis is highly nontrivial and technical. We discover a pivotal inequality for delineating the convexity and concavity of the iterative function and establish theories to guarantee the PCP property and convergence. We also develop theories to determine a computable initial guess within a theoretical "safe" interval. Intriguingly, we find that the unique positive root of a cubic polynomial always falls within this interval. Our PCP NR method is versatile and can be seamlessly integrated into any RMHD scheme that requires the recovery of primitive variables, potentially leading to a broad impact in this field. As an application, we incorporate it into a discontinuous Galerkin method, resulting in fully PCP schemes. Several numerical experiments demonstrate the efficiency and robustness of the PCP NR method.

math.NA↗

arXiv2024

Bound-Preserving Framework for Central-Upwind Schemes for General Hyperbolic Conservation Laws

Central-upwind (CU) schemes are Riemann-problem-solver-free finite-volume methods widely applied to a variety of hyperbolic systems of PDEs. Exact solutions of these systems typically satisfy certain bounds, and it is highly desirable or even crucial for the numerical schemes to preserve these bounds. In this paper, we develop and analyze bound-preserving (BP) CU schemes for general hyperbolic systems of conservation laws. Unlike many other Godunov-type methods, CU schemes cannot, in general, be recast as convex combinations of first-order BP schemes. Consequently, standard BP analysis techniques are invalidated. We address these challenges by establishing a novel framework for analyzing the BP property of CU schemes. To this end, we discover that the CU schemes can be decomposed as a convex combination of several intermediate solution states. Thanks to this key finding, the goal of designing BPCU schemes is simplified to the enforcement of four more accessible BP conditions, each of which can be achieved with the help of a minor modification of the CU schemes. We employ the proposed approach to construct provably BPCU schemes for the Euler equations of gas dynamics. The robustness and effectiveness of the BPCU schemes are validated by several demanding numerical examples, including high-speed jet problems, flow past a forward-facing step, and a shock diffraction problem.

math.NA↗

arXiv2024

GQL-Based Bound-Preserving and Locally Divergence-Free Central Discontinuous Galerkin Schemes for Relativistic Magnetohydrodynamics

This paper develops novel and robust central discontinuous Galerkin (CDG) schemes of arbitrarily high-order accuracy for special relativistic magnetohydrodynamics (RMHD) with a general equation of state (EOS). These schemes are provably bound-preserving (BP), i.e., consistently preserve the upper bound for subluminal fluid velocity and the positivity of density and pressure, while also (locally) maintaining the divergence-free (DF) constraint for the magnetic field. For 1D RMHD, the standard CDG method is exactly DF, and its BP property is proven under a condition achievable by BP limiter. For 2D RMHD, we design provably BP and locally DF CDG schemes based on the suitable discretization of a modified RMHD system. A key novelty in our schemes is the discretization of additional source terms in the modified RMHD equations, so as to precisely counteract the influence of divergence errors on the BP property across overlapping meshes. We provide rigorous proofs of the BP property for our CDG schemes and first establish the theoretical connection between BP and discrete DF properties on overlapping meshes for RMHD. Owing to the absence of explicit expressions for primitive variables in terms of conserved variables, the constraints of physical bounds are strongly nonlinear, making the BP proofs highly nontrivial. We overcome these challenges through technical estimates within the geometric quasilinearization (GQL) framework, which converts the nonlinear constraints into linear ones. Furthermore, we introduce a new 2D cell average decomposition on overlapping meshes, which relaxes the theoretical BP CFL constraint and reduces the number of internal nodes, thereby enhancing the efficiency of the 2D BP CDG method. We implement the proposed CDG schemes for extensive RMHD problems with various EOSs, demonstrating their robustness and effectiveness in challenging scenarios.

math.NA↗

arXiv2024

High-order accurate positivity-preserving and well-balanced discontinuous Galerkin schemes for ten-moment Gaussian closure equations with source terms

This paper proposes novel high-order accurate discontinuous Galerkin (DG) schemes for the one- and two-dimensional ten-moment Gaussian closure equations with source terms defined by a known potential function. Our DG schemes exhibit the desirable capability of being well-balanced (WB) for a known hydrostatic equilibrium state while simultaneously preserving positive density and positive-definite anisotropic pressure tensor. The well-balancedness is built on carefully modifying the solution states in the Harten-Lax-van Leer-contact (HLLC) flux, and appropriate reformulation and discretization of the source terms. Our novel modification technique overcomes the difficulties posed by the anisotropic effects, maintains the high-order accuracy, and ensures that the modified solution state remains within the physically admissible state set. Positivity-preserving analyses of our WB DG schemes are conducted by using several key properties of the admissible state set, the HLLC flux and the HLLC solver, as well as the geometric quasilinearization (GQL) approach in [Wu & Shu, SIAM Review, 65: 1031-1073, 2023], which was originally applied to analyze the admissible state set and physical-constraints-preserving schemes for the relativistic magnetohydrodynamics in [Wu & Tang, M3AS, 27: 1871-1928, 2017], to address the difficulties arising from the nonlinear constraints on pressure tensor. Moreover, the proposed WB DG schemes satisfy the weak positivity for the cell averages, implying the use of a scaling limiter to enforce the physical admissibility of the DG solution polynomials at certain points of interest. Extensive numerical experiments are conducted to validate the preservation of equilibrium states, accuracy in capturing small perturbations to such states, robustness in solving problems involving low density or low pressure, and high resolution for both smooth and discontinuous solutions.

math.NA↗

arXiv2023

Critical Sampling for Robust Evolution Operator Learning of Unknown Dynamical Systems

Given an unknown dynamical system, what is the minimum number of samples needed for effective learning of its governing laws and accurate prediction of its future evolution behavior, and how to select these critical samples? In this work, we propose to explore this problem based on a design approach. Starting from a small initial set of samples, we adaptively discover critical samples to achieve increasingly accurate learning of the system evolution. One central challenge here is that we do not know the network modeling error since the ground-truth system state is unknown, which is however needed for critical sampling. To address this challenge, we introduce a multi-step reciprocal prediction network where forward and backward evolution networks are designed to learn the temporal evolution behavior in the forward and backward time directions, respectively. Very interestingly, we find that the desired network modeling error is highly correlated with the multi-step reciprocal prediction error, which can be directly computed from the current system state. This allows us to perform a dynamic selection of critical samples from regions with high network modeling errors for dynamical systems. Additionally, a joint spatial-temporal evolution network is introduced which incorporates spatial dynamics modeling into the temporal evolution prediction for robust learning of the system evolution operator with few samples. Our extensive experimental results demonstrate that our proposed method is able to dramatically reduce the number of samples needed for effective learning and accurate prediction of evolution behaviors of unknown dynamical systems by up to hundreds of times.

cs.LG↗