arXiv subjects

Kailiang Wu

Publications and source records attributed to Kailiang Wu.

At least 19 recordsLinked to original sources

arXiv2026

Fully Discrete Multi-Entropy Stability of High-Order Schemes for Compressible MHD: A Weak-to-Strong Framework

We establish a fully discrete weak-to-strong (W2S) multi-entropy stability theory for arbitrarily high-order finite-volume and discontinuous Galerkin (DG) approximations of the ideal compressible magnetohydrodynamics (MHD) equations on general polytopal meshes, whereby a single numerical update simultaneously satisfies discrete entropy inequalities for any prescribed finite family of convex Harten entropy pairs. Because physical entropies are defined only for positive density and pressure, the central analytical difficulty lies in reconciling discrete entropy stability with positivity preservation, the magnetic divergence constraint, and the nonconservative Godunov--Powell coupling. Building upon the provably positivity-preserving MHD framework of Wu and Shu, we develop a weak multi-entropy analysis of the common cell-average evolution of finite-volume and DG methods. Through convex decomposition and relative entropy, we establish two sufficient criteria for this weak stability general polytopal meshes. Compatible magnetic corrections and locally divergence-free approximations cooperatively offset the nonconservative Godunov--Powell coupling and provide the discrete cancellations essential to both positivity-preservation and entropy stability. A W2S lifting then unifies positivity and entropy limiting into a single cellwise scaling limiter, achieving strong multi-entropy stability while strictly preserving cell averages and the locally divergence-free magnetic field. High-order temporal accuracy follows from variable-step strong-stability-preserving multistep methods. Numerical experiments confirm the theoretical findings and demonstrate computational robustness. This framework provides a rigorous, fully discrete foundation for unifying physical admissibility, magnetic divergence control, and multi-entropy stability in high-order MHD approximations.

math.NA↗

arXiv2026

Structure of ENO Entropy Dissipation: Parity Dichotomy for the ENO--TV Conjecture and Shift Cohomology

Essentially non-oscillatory (ENO) reconstruction provides a key mechanism for designing high-order entropy-stable schemes for hyperbolic conservation laws, with its sign property ensuring nonnegative local dissipation for a prescribed entropy. However, two fundamental questions concerning convergence remain open: whether this dissipation provides the coercivity required for weak-BV compactness, as posited by the ENO--TV conjecture, and whether entropy stability transfers from the prescribed entropy pair to additional pairs. This paper resolves the ENO--TV conjecture by establishing a sharp parity dichotomy: it holds if and only if the reconstruction order $k=2$ or $k$ is odd, and fails for all even orders $k\ge 4$.The key to our proof is a localization principle that eliminates dependence on nonlinear adaptive stencil selection, establishing a two-sided equivalence between ENO dissipation and a canonical finite-difference functional. For odd orders, the conjecture is proved via a hidden quadratic energy and novel discrete Gagliardo--Nirenberg inequalities. For even orders $k\ge 4$, ENO null modes, on which ENO dissipation vanishes, yield counterexamples that disprove the conjecture. This dichotomy extends to quasi-uniform meshes, but for every $k\ge 2$, the conjecture can fail on non-quasi-uniform meshes. Addressing the above second open question, we discover on ENO null modes that local entropy transfer is governed by the first cohomology of a unipotent shift. Using apolar duality and binary covariants, we compute the dimensions of the associated cohomology subspaces and prove that smooth local entropy transfer encounters generic obstructions for every $k\ge 4$. By revealing how ENO null modes link global coercivity and local entropy compatibility, this work provides a structural foundation for the compactness and convergence analysis of high-order entropy-stable discretizations.

math.AP↗

arXiv2026

Invariant domain preservation for hybrid point-value and cell-average discretizations of hyperbolic equations on general meshes

This paper presents a unified invariant-domain-preserving (IDP) framework for hybrid discretizations of hyperbolic conservation laws, including active flux and PAMPA methods, in which cell averages are updated conservatively while cell-boundary point values evolve under a possibly non-conservative operator. The main challenge is to preserve admissibility for these two coupled sets of states under a single local stability condition, without adding evolved degrees of freedom or relying on post-update repairs. For the point-value update, we introduce an admissibility transform based on a barrier--Legendre map for convex admissible interiors described by concave constraints, and prove that the inverse map is globally defined and Lipschitz continuous on the relevant sets. For the conservative cell-average update, we establish a structural obstruction theorem: the single-state continuous physical flux built from admissible boundary traces alone cannot provide a conservative IDP guarantee, for any prescribed CFL number, when internal reconstruction values are uncontrolled. This identifies the missing local control that must be supplied by an additional IDP flux mechanism. To realize this mechanism explicitly, we combine cell average decompositions (CAD), geometric quasilinearization, and local a priori scaling to construct admissible, generally discontinuous trace states before flux evaluation. Under an explicit trace-based CFL condition, with constants determined by local CAD weights and trace-state wave-speed bounds, the coupled hybrid update preserves the prescribed invariant domain. Concrete third-order schemes are developed on triangular, Cartesian, convex quadrilateral, and general convex polygonal meshes. Numerical results demonstrate the designed order of accuracy for smooth solutions and the strict preservation of physical admissibility.

math.NA↗

arXiv2026

Runge--Kutta-Aligned Oscillation Elimination: Restoring Superconvergence for Fully Discrete Shock-Capturing DG Schemes

Nonlinear stabilization is indispensable for discontinuous Galerkin (DG) discretizations of hyperbolic conservation laws, yet it typically disrupts the delicate error structure required for superconvergence analysis. Consequently, existing theory has largely been restricted to linear or semi-discrete schemes lacking oscillation control. This paper bridges this gap by proposing a Runge--Kutta (RK) aligned oscillation-eliminating (OE) DG framework that restores the superconvergence properties. By synchronizing the pseudo-time step in the OE procedure with the cumulative RK stage coefficients, we unlock a cancellation mechanism for low-order interface errors that is inaccessible to standard OEDG formulations. We rigorously prove that this fully discrete scheme achieves $(k+2)$-th order superconvergence to a tailored projection of the exact solution for linear conservation laws in both one and two dimensions, while maintaining the non-oscillatory shock-capturing capabilities of the original method. Moreover, we establish a general guiding principle for designing a class of OE-type DG schemes that exhibit such superconvergence. Key theoretical innovations include the construction of stage-aligned correction functions to compensate for nonlinear OE sources and the discovery of a two-dimensional projection operator that preserves outflow-edge averages, a property essential to close the discrete shift estimates. Numerical experiments confirm the predicted superconvergence rates and demonstrate that RK alignment preserves the parameter-free robustness of OEDG for problems with strong discontinuities.

math.NA↗

arXiv2026

On Energy Laws and Stability of First-Subdiagonal Pade Approximants for Linear Seminegative Problems

We derive an explicit discrete energy identity for rational time discretizations generated by the first-subdiagonal Padé approximants of the exponential for solving linear seminegative problems. This work extends the diagonal Padé energy laws in [Z. Sun, Y. Wei, and K. Wu, SIAM J. Numer. Anal., 60 (2022)] to the first-subdiagonal family. The main new ingredient is an explicit Cholesky-type factorization of the energy coefficient matrix associated with the semi-inner-product terms in the discrete energy identity. The construction and proof of this factorization are nontrivial, since the matrix entries are alternating sums of Padé coefficients and the triangular factor has a parity-dependent factorial structure. We prove the factorization by reducing it to scalar rational identities and establishing them through finite product reductions and telescoping summations. Together with a β-coefficient cancellation, the factorization yields an exact discrete energy law that recovers the classical unconditional contractivity for linear seminegative problems. Numerical experiments adapted from the diagonal Padé energy-law setting illustrate the predicted order and verify the discrete dissipation identity.

math.NA↗

arXiv2026

High-order fully discrete multi-entropy-stable and bound-preserving schemes for relativistic Euler equations

A discrete entropy inequality is the principal nonlinear stability estimate available for systems of conservation laws, and evaluating it presupposes a physically admissible state. So far, however, the two have been secured separately. Entropy-stable schemes are almost always semi-discrete, are built around one selected entropy pair, and take for granted the positivity of density and pressure that makes the entropy well defined in the first place, whereas bound-preserving limiters keep the solution admissible but deliver no entropy estimate. For the special relativistic Euler equations, the two cannot be separated at all, since the conservative-to-primitive map is implicit, and an inadmissible state therefore has no entropy to correct. Here we construct high-order discontinuous Galerkin and finite volume schemes that, to our knowledge, for the first time, are entropy stable in the fully discrete sense for an arbitrary prescribed finite family of convex entropy pairs, a property we call multi-entropy stability, and are provably admissible wherever an entropy is evaluated. All of this is achieved by a single cellwise projection, and neither conservation nor the design order is lost. The construction rests on relativistic causality, which bounds every characteristic speed by the speed of light. Consequently, the numerical viscosity can be fixed once for all states and all equations of state, and one two-point building block then serves the whole entropy family. Since only the convexity of the admissible set and this speed bound are used, the same route remains open for related systems. Finally, in computations with four equations of state, the schemes retain high-order accuracy close to vacuum, produce no inadmissible state in strong shocks, near-vacuum shock--vortex interaction or jets with Lorentz factor above $70$, and confirm the monotone decay of every enforced discrete entropy.

math.NA↗

arXiv2026

Compact Finite-Average-Moment Schemes with Single-Step Oscillation Elimination for Hyperbolic Conservation Laws

High-order shock-capturing schemes often face severe computational bottlenecks due to wide stencils, stagewise nonlinear weights, and expensive local characteristic decompositions. To address this, we propose a compact finite-average-moment (FAM) framework for hyperbolic conservation laws that completely decouples the formal spatial order from the number of evolved local degrees of freedom. By evolving only a $\mathbb{P}^1$ moment state (i.e., cell averages and scaled first-order moments), we achieve up to sixth-order accuracy ($k=3, 4, 5, 6$) via compact, polynomially exact \emph{linear} moment reconstructions. A key algorithmic innovation is consolidating the nonlinear stabilization into a single, derivative-free oscillation-elimination (OE) procedure applied only after the final Runge--Kutta stage. This OE procedure exactly preserves cell averages while applying an explicit exponential correction to the first-order moments, avoiding repeated stagewise nonlinear weights or limiting. Fourier analysis of the linear backbone demonstrates $k$th-order accuracy, strong spurious mode damping, and $(k+1)$th-order cell-average superconvergence. Extensive numerical experiments on scalar laws and the Euler equations verify the expected smooth accuracy and robust, nonoscillatory shock resolution. Notably, by performing componentwise reconstruction directly in conservative variables and streamlining stabilization, the FAM schemes deliver significantly lower complete-run wall-clock times compared to the multi-resolution weighted essentially non-oscillatory (MR-WENO), unified-stencil Hermite WENO (HWENO-U), and OE-HWENO methods.

math.NA↗

arXiv2026

Hidden Accuracy and Superconvergence Analysis of Central Discontinuous Galerkin Methods on Overlapping Meshes

This paper establishes the first rigorous superconvergence theory for semidiscrete and fully discrete central discontinuous Galerkin (CDG) methods for linear hyperbolic equations on overlapping meshes. While the optimal $L^2$ convergence of $\mathbb{Q}^k$ CDG schemes was established on uniform Cartesian meshes by Liu, Shu, and Zhang [ SIAM J. Numer. Anal.}, 56 (2018), pp. 520--541], their observed $\mathcal{O}(h^{k+2})$ pointwise superconvergence has remained unproven, due to the loss of standard single-mesh Galerkin orthogonality inherent in the CDG overlapping structure. To overcome this fundamental barrier, we introduce a projection-correction framework that identifies a hidden superconvergent mechanism: an asymptotic weak residual cancellation in one dimension, and a high-order cancellation-by-aggregation (HOCA) mechanism in multiple dimensions. This HOCA approach overcomes the analytical challenge posed by coupled primal-dual directional residuals, recovering critical error cancellation properties absent from the standard variational formulation. Consequently, we provide the rigorous proof of the conjectured $\mathcal{O}(h^{k+2})$ pointwise superconvergence in the discrete $\ell^{\infty}$ norm across all superconvergent points. Furthermore, we reveal that under a systematically corrected initialization, this framework yields a previously undiscovered, stronger cell-average superconvergence estimate of order $\mathcal{O}(h^{\min\{2k+1,k+3\}})$. The theory is extended to fully discrete explicit Runge--Kutta CDG schemes, where stagewise corrected errors are constructed to preserve spatial superconvergence up to temporal truncation errors, yielding a stable reconstruction-based postprocessing estimate. Numerical experiments in one and two spatial dimensions confirm the sharpness of the theoretical rates.

math.NA↗

arXiv2026

GQL-Based Physical-Constraint-Preserving High-Order Finite Difference Schemes for Special Relativistic Hydrodynamics in Arbitrary Dimensions

High-order accurate simulations of special relativistic hydrodynamics (RHD) are prone to numerical breakdown if intrinsic physical constraints (positive rest-mass density/pressure and subluminal velocity) are violated near strong discontinuities. In this work, we develop a robust and efficient physical-constraint-preserving (PCP) flux-limiting framework for high-order schemes, using finite-difference WENO as a representative example. By leveraging the geometric quasilinearization (GQL) representation, which equivalently reformulates the nonlinear RHD constraints into a family of linear inequalities, we integrate a Zalesak-type Flux-Corrected Transport (FCT) update into a scalar-style limiter that acts directly on conservative variables. A critical innovation is the explicit, non-iterative determination of limiting parameters via a rational stereographic parameterization of the GQL normal vector. This technique transforms the required worst-case minimization over auxiliary variables into a generalized Rayleigh-quotient formulation, allowing the optimal parameters to be obtained by solving small symmetric eigenvalue problems ($2\times2$ in 1D; $(d+1)\times(d+1)$ in $d$ dimensions). Relaxed variants are further introduced to reduce computational costs in multidimensions while retaining the PCP guarantee. Extensive numerical benchmarks ranging from 1D to 3D, including ultra-relativistic Riemann problems and astrophysical jets, demonstrate that the proposed method robustly enforces physical admissibility, sharply resolves discontinuities, and maintains design-order accuracy for smooth solutions.

math.NA↗

arXiv2026

EPO: A Unified Framework for Entropy Stability, Positivity, and Oscillation Suppression

High-order finite volume and discontinuous Galerkin methods are often stabilized by separate nonlinear devices for admissibility, entropy control, and oscillation suppression. This separation hides a simple geometric fact: all three act on the same cellwise candidate state. We propose a general framework (termed EPO) unifying fully discrete entropy stability, positivity/bound preservation, and spurious oscillation elimination. Starting from a candidate update, we scale along the ray anchored at its updated cell average. The admissible-state constraint, the entropy constraint, and the oscillation-suppressing constraint each define an admissibility radius on that ray, and the applied limiter is their minimum. The decisive analytical ingredient is a {\em weak entropy stability} at the level of the updated cell average. A two-point Lax--Friedrichs/Riemann-average entropy inequality yields local cell-average entropy budgets, and the same radial scaling mechanism behind Zhang--Shu positivity preservation lifts these weak budgets to strong quadrature-based entropy inequalities. The framework is therefore not a summation-by-parts, split-form, or flux-differencing construction: EPO acts on a candidate finite volume or discontinuous Galerkin update and converts weak average information into fully discrete nodal entropy stability. {\em The construction also works for any prescribed finite family of convex entropy pairs. Each pair yields its own entropy radius, and taking the minimum enforces fully discrete entropy stability for all of them simultaneously.} We prove the preservation of cell averages, invariant-set preservation, local and global strong entropy inequalities, stagewise budgets for strong-stability-preserving (SSP) Runge--Kutta methods, an SSP multistep variant that retains the designed high-order temporal accuracy, and extensions on rectangular and unstructured triangular meshes.

math.NA↗

arXiv2025

High Order Numerical Methods Preserving Invariant Domain for Hyperbolic and Related Systems

Admissible states in hyperbolic systems and related equations often form a convex invariant domain. Numerical violations of this domain can lead to loss of hyperbolicity, resulting in illposedness and severe numerical instabilities. It is therefore crucial for numerical schemes to preserve the invariant domain to ensure both physically meaningful solutions and robust computations. For complex systems, constructing invariant-domain-preserving (IDP) schemes is highly nontrivial and particularly challenging for high-order accurate methods. This paper presents a comprehensive survey of IDP schemes for hyperbolic and related systems, with a focus on the most popular approaches for constructing provable IDP schemes. We first give a systematic review of the fundamental approaches for establishing the IDP property in first-order accurate schemes, covering finite difference, finite volume, finite element, and residual distribution methods. Then we focus on two widely used and actively developed classes of high order IDP schemes as well as their recent developments, most of which have emerged in the past decade. The first class of methods seeks an intrinsic weak IDP property in high-order schemes and then designs polynomial limiters to enforce a strong IDP property at the points of interest. This generic approach applies to high-order finite volume and discontinuousGalerkin schemes. The second class is based on the flux limiting approaches, which originated from the flux-corrected transport method and can be adapted to a broader range of spatial discretizations, including finite difference and continuous finite element methods. In this survey, we elucidate the main ideas in the construction of IDP schemes, provide some new perspectives and insights, with extensive examples, and numerical experiments in gas dynamics and magnetohydrodynamics.

math.NA↗

arXiv2025

Constraint-Preserving High-Order Compact OEDG Method for Spherically Symmetric Einstein-Euler System

Numerical simulation of the spherically symmetric Einstein--Euler (EE) system faces severe challenges due to the stringent physical admissibility constraints of relativistic fluids and the geometric singularities inherent in metric evolution. This paper proposes a high-order Constraint-Preserving (CP) compact Oscillation-Eliminating Discontinuous Galerkin (cOEDG) method specifically tailored to address these difficulties. The method integrates a scale-invariant oscillation-eliminating mechanism [M. Peng, Z. Sun, K. Wu, Math. Comp., 94: 1147--1198, 2025] into a compact Runge--Kutta DG framework. By characterizing the convex invariant region of the hydrodynamic subsystem with general barotropic equations of state, we prove that the proposed scheme preserves physical realizability (specifically, positive density and subluminal velocity) directly in terms of conservative variables, thereby eliminating the need for complex primitive-variable checks. To ensure the geometric validity of the spacetime, we introduce a bijective transformation of the metric potentials. Rather than evolving the constrained metric components directly, the scheme advances unconstrained auxiliary variables whose inverse mapping automatically enforces strict positivity and asymptotic bounds without any limiters. Combined with a compatible high-order boundary treatment, the resulting CPcOEDG method exhibits robust stability and design-order accuracy in capturing strong gravity-fluid interactions, as demonstrated by simulations of black hole accretion and relativistic shock waves.

math.NA↗

arXiv2025

An active-flux-type scheme for ideal MHD with provable positivity and discrete divergence-free property

We develop a positivity-preserving (PP) PAMPA (Point-Average-Moment PolynomiAl-interpreted) scheme that enforces a discrete divergence-free (DDF) magnetic field for ideal MHD on Cartesian grids. Extending our 1D invariant-domain-preserving (IDP) PAMPA framework (Abgrall, Jiao, Liu, Wu, SIAM J. Sci. Comput., to appear) to multidimensional, multiwave MHD, the method combines a limiter-free PP update of interface point values via a new nonconservative reformulation with a local DDF projection. Cell averages are provably PP under a mild a~priori positivity condition on one cell-centered state, using: (i) DDF-constrained interface values, (ii) a PP limiter only at the cell center, (iii) a PP flux with appropriate wave-speed bounds, and (iv) a suitable discretization of the Godunov--Powell source term. The PP proof employs geometric quasi-linearization (GQL; Wu & Shu, SIAM Review, 2023), which linearizes the pressure constraint. The scheme avoids explicit polynomial reconstructions, is compatible with arbitrarily high-order strong-stability-preserving (SSP) time integration, and is simple to implement. Robustness and resolution are enhanced by a problem-independent Lax-type entropy troubled-cell indicator using only two characteristic speeds and a convex oscillation elimination (COE) mechanism with a new intercell-difference norm. Tests -- including a blast wave with plasma $β\approx 2.51\times 10^{-6}$ and jets up to Mach $10^{4}$ -- show high-order accuracy, sharp MHD-structure resolution, and strong-shock robustness. To our knowledge, this is the first active-flux-type ideal-MHD method rigorously PP for both cell averages and interface point values while maintaining DDF throughout.

math.NA↗

arXiv2025

Provably realizability-preserving finite volume method for quadrature-based moment models of kinetic equations

Quadrature-based moment methods (QBMM) provide tractable closures for multiscale kinetic equations, with diverse applications across aerosols, sprays, and particulate flows, etc. However, for the derived hyperbolic moment-closure systems, seeking numerical schemes preserving moment realizability is essential yet challenging due to strong nonlinear coupling and the lack of explicit conservative-to-flux maps. This paper proposes and analyzes a provably realizability-preserving finite-volume method for five-moment systems closed by the two-node Gaussian-EQMOM and three-point HyQMOM. Rather than relying on kinetic fluxes, we recast the realizability condition into a nonnegative quadratic form in the moment vector, reducing the original nonlinear constraints to bilinear inequalities amenable to analysis. On this basis, we construct a tailored Harten--Lax--van Leer (HLL) flux with rigorously derived wave speeds and intermediate states that embed realizability directly into the flux evaluation. We prove sufficient realizability-preserving conditions under explicit Courant--Friedrichs--Lewy (CFL) constraints in the collisionless case, and for BGK relaxation, we obtain coupled time-step conditions involving a realizability radius; a semi-implicit BGK variant inherits the collisionless CFL. From a multiscale perspective, the analysis yields stability conditions uniform in the relaxation time and supports stiff-to-kinetic transitions. A practical limiter enforces strict realizability of reconstructed interface states without degrading accuracy. Numerical experiments demonstrate the accuracy, robustness in low-density regions, and realizability for both closures. This framework unifies realizability preservation for solving hyperbolic moment systems with complex closures and extends naturally to higher-order space--time discretizations.

math.NA↗

arXiv2025

Bound-Preserving WENO Schemes for Temple-class systems

This paper explores numerical schemes for Temple-class systems, which are integral to various applications including one-dimensional two-phase flow, elasticity, traffic flow, and sedimentation. Temple-class systems are characterized by conservative equations, with different pressure function expressions leading to specific models such as the Aw-Rascle-Zhang (ARZ) traffic model and the sedimentation model. Our work extends existing studies by introducing a moving mesh approach to address the challenges of preserving non-convex invariant domains, a common issue in the numerical simulation of such systems. Our study outlines a novel bound-preserving (BP) and conservative numerical scheme, designed specifically for non-convex sets in Temple-class systems, which is critical for avoiding non-physical solutions and ensuring robustness in simulations. We develop both local and global BP methods based on finite difference schemes, with numerical experiments demonstrating the effectiveness and reliability of our methods. Furthermore, a parameterized flux limiter is introduced to restrict high-order fluxes and maintain bound preservation. This innovation marks the first time such a parameterized approach has been applied to non-convex sets, offering significant improvements over traditional methods. The findings presented extend beyond theoretical implications, as they are applicable to general Temple-class systems and can be tailored to ARZ traffic flow networks, highlighting the versatility and broad applicability of our approach. The paper contributes significantly to the field by providing a comprehensive method that maintains the physical and mathematical constrains of Temple-class systems.

math.NA↗

arXiv2025

Provably positivity-preserving, globally divergence-free central DG methods for ideal MHD system

This paper proposes a numerical method, termed PosDiv-CDG, that provably preserves both positivity and the globally divergence-free (DF) condition at arbitrarily high order in multiple dimensions. It resolves the fundamental structural incompatibility between standard positivity-preserving limiters and global DF enforcement in the central discontinuous Galerkin (CDG) framework. The method integrates a novel positivity-limiting strategy, a modified dissipation mechanism guided by convex decomposition, and an auxiliary evolution equation for the magnetic field, which are designed based on rigorous theoretical analysis. Notably, we provide a rigorous proof of positivity preservation for the updated cell averages under an explicit CFL-type condition. The proof leverages the geometric quasi-linearization (GQL) technique, which reformulates the nonlinear positivity constraint into an equivalent linear form. This enables the derivation of flux-based inequalities and technical estimates under the global DF constraint. To suppress nonphysical oscillations near shocks, we develop a compact, non-intrusive convex-oscillation-suppressing (COS) procedure based on the entropy function. The COS process acts only on non-magnetic variables, avoids costly characteristic decomposition, and maintains both the globally DF property and high-order accuracy. Several challenging experiments -- including low plasma-beta MHD jets with Mach numbers up to 1,000,000 -- demonstrate the proposed method robustness, high-order accuracy, non-oscillatory behavior, and its ability to preserve both positivity and globally DF structures under extreme conditions.

math.NA↗

arXiv2025

Bound preserving Point-Average-Moment PolynomiAl-interpreted (PAMPA) scheme: one-dimensional case

We propose a bound-preserving (BP) Point-Average-Moment PolynomiAl-interpreted (PAMPA) scheme by blending third-order and first-order constructions. The originality of the present construction is that it does not need any explicit reconstruction within each element, and therefore the construction is very flexible. The scheme employs a classical blending approach between a first-order BP scheme and a high-order scheme that does not inherently preserve bounds. The proposed BP PAMPA scheme demonstrates effectiveness across a range of problems, from scalar cases to systems such as the Euler equations of gas dynamics. We derive optimal blending parameters for both scalar and system cases, with the latter based on the recent geometric quasi-linearization (GQL) framework of [Wu \& Shu, {\em SIAM Review}, 65 (2023), pp. 1031--1073]. This yields explicit, optimal blending coefficients that ensure positivity and control spurious oscillations in both point values and cell averages. This framework incorporates a convex blending of fluxes and residuals from both high-order and first-order updates, facilitating a rigorous BP property analysis. Sufficient conditions for the BP property are established, ensuring robustness while preserving high-order accuracy. Numerical tests confirm the effectiveness of the BP PAMPA scheme on several challenging problems.

math.NA↗

arXiv2025

DUE: A Deep Learning Framework and Library for Modeling Unknown Equations

Equations, particularly differential equations, are fundamental for understanding natural phenomena and predicting complex dynamics across various scientific and engineering disciplines. However, the governing equations for many complex systems remain unknown due to intricate underlying mechanisms. Recent advancements in machine learning and data science offer a new paradigm for modeling unknown equations from measurement or simulation data. This paradigm shift, known as data-driven discovery or modeling, stands at the forefront of AI for science, with significant progress made in recent years. In this paper, we introduce a systematic framework for data-driven modeling of unknown equations using deep learning. This versatile framework is capable of learning unknown ODEs, PDEs, DAEs, IDEs, SDEs, reduced or partially observed systems, and non-autonomous differential equations. Based on this framework, we have developed Deep Unknown Equations (DUE), an open-source software package designed to facilitate the data-driven modeling of unknown equations using modern deep learning techniques. DUE serves as an educational tool for classroom instruction, enabling students and newcomers to gain hands-on experience with differential equations, data-driven modeling, and contemporary deep learning approaches such as FNN, ResNet, generalized ResNet, operator semigroup networks (OSG-Net), and Transformers. Additionally, DUE is a versatile and accessible toolkit for researchers across various scientific and engineering fields. It is applicable not only for learning unknown equations from data but also for surrogate modeling of known, yet complex, equations that are costly to solve using traditional numerical methods. We provide detailed descriptions of DUE and demonstrate its capabilities through diverse examples, which serve as templates that can be easily adapted for other applications.

cs.LG↗