On enforcing non-negativity in polynomial approximations in high dimensions

Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negative (resp. non-positive), or bounded within a given range, due to constraints posed by the underlying physical problems. Efficient numerical methods are thus needed to enforce such conditions. In this paper, we discuss effective numerical algorithms for polynomial approximation under non-negativity constraints. We first formulate the constrained optimization problem, its primal and dual forms, and then discuss efficient first-order convex optimization methods, with a particular focus on high dimensional problems. Numerical examples are provided, for up to $200$ dimensions, to demonstrate the effectiveness and scalability of the methods

Is the Classic Convex Decomposition Optimal for Bound-Preserving Schemes in Multiple Dimensions?

Since proposed in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229: 3091--3120, 2010], the Zhang--Shu framework has attracted extensive attention and motivated many bound-preserving (BP) high-order discontinuous Galerkin and finite volume schemes for various hyperbolic equations. A key ingredient in the framework is the decomposition of the cell averages of the numerical solution into a convex combination of the solution values at certain quadrature points, which helps to rewrite high-order schemes as convex combinations of formally first-order schemes. The classic convex decomposition originally proposed by Zhang and Shu has been widely used over the past decade. It was verified, only for the 1D quadratic and cubic polynomial spaces, that the classic decomposition is optimal in the sense of achieving the mildest BP CFL condition. Yet, it remained unclear whether the classic decomposition is optimal in multiple dimensions. In this paper, we find that the classic multidimensional decomposition based on the tensor product of Gauss--Lobatto and Gauss quadratures is generally not optimal, and we discover a novel alternative decomposition for the 2D and 3D polynomial spaces of total degree up to 2 and 3, respectively, on Cartesian meshes. Our new decomposition allows a larger BP time step-size than the classic one, and moreover, it is rigorously proved to be optimal to attain the mildest BP CFL condition, yet requires much fewer nodes. The discovery of such an optimal convex decomposition is highly nontrivial yet meaningful, as it may lead to an improvement of high-order BP schemes for a large class of hyperbolic or convection-dominated equations, at the cost of only a slight and local modification to the implementation code. Several numerical examples are provided to further validate the advantages of using our optimal decomposition over the classic one in terms of efficiency

A Flux-Differencing Formula for Split-Form Summation By Parts Discretizations of Non-Conservative Systems: Applications to Subcell Limiting for magneto-hydrodynamics

In this paper, we show that diagonal-norm summation by parts (SBP) discretizations of general non-conservative systems of hyperbolic balance laws can be rewritten as a finite-volume-type formula, also known as flux-differencing formula, if the non-conservative terms can be written as the product of a local and a symmetric contribution. Furthermore, we show that the existence of a flux-differencing formula enables the use of recent subcell limiting strategies to improve the robustness of the high-order discretizations. To demonstrate the utility of the novel flux-differencing formula, we construct hybrid schemes that combine high-order SBP methods (the discontinuous Galerkin spectral element method and a high-order SBP finite difference method) with a compatible low-order finite volume (FV) scheme at the subcell level. We apply the hybrid schemes to solve challenging magnetohydrodynamics (MHD) problems featuring strong shocks

A new discretely divergence-free positivity-preserving high-order finite volume method for ideal MHD equations

This paper proposes and analyzes a novel efficient high-order finite volume method for the ideal magnetohydrodynamics (MHD). As a distinctive feature, the method simultaneously preserves a discretely divergence-free (DDF) constraint on the magnetic field and the positivity-preserving (PP) property, which ensures the positivity of density, pressure, and internal energy. To enforce the DDF condition, we design a new discrete projection approach that projects the reconstructed point values at the cell interface into a DDF space, without using any approximation polynomials. This projection method is highly efficient, easy to implement, and particularly suitable for standard high-order finite volume WENO methods, which typically return only the point values in the reconstruction. Moreover, we also develop a new finite volume framework for constructing provably PP schemes for the ideal MHD system. The framework comprises the discrete projection technique, a suitable approximation to the Godunov--Powell source terms, and a simple PP limiter. We provide rigorous analysis of the PP property of the proposed finite volume method, demonstrating that the DDF condition and the proper approximation to the source terms eliminate the impact of magnetic divergence terms on the PP property. The analysis is challenging due to the internal energy function's nonlinearity and the intricate relationship between the DDF and PP properties. To address these challenges, the recently developed geometric quasilinearization approach is adopted, which transforms a nonlinear constraint into a family of linear constraints. Finally, we validate the effectiveness of the proposed method through several benchmark and demanding numerical examples. The results demonstrate that the proposed method is robust, accurate, and highly effective, confirming the significance of the proposed DDF projection and PP techniques.Comment: 26 page

A Physical-Constraint-Preserving Finite Volume WENO Method for Special Relativistic Hydrodynamics on Unstructured Meshes

This paper presents a highly robust third-order accurate finite volume weighted essentially non-oscillatory (WENO) method for special relativistic hydrodynamics on unstructured triangular meshes. We rigorously prove that the proposed method is physical-constraint-preserving (PCP), namely, always preserves the positivity of the pressure and the rest-mass density as well as the subluminal constraint on the fluid velocity. The method is built on a highly efficient compact WENO reconstruction on unstructured meshes, a simple PCP limiter, the provably PCP property of the Harten--Lax--van Leer flux, and third-order strong-stability-preserving time discretization. Due to the relativistic effects, the primitive variables (namely, the rest-mass density, velocity, and pressure) are highly nonlinear implicit functions in terms of the conservative variables, making the design and analysis of our method nontrivial. To address the difficulties arising from the strong nonlinearity, we adopt a novel quasilinear technique for the theoretical proof of the PCP property. Three provable convergence-guaranteed iterative algorithms are also introduced for the robust recovery of primitive quantities from admissible conservative variables. We also propose a slight modification to an existing WENO reconstruction to ensure the scaling invariance of the nonlinear weights and thus to accommodate the homogeneity of the evolution operator, leading to the advantages of the modified WENO reconstruction in resolving multi-scale wave structures. Extensive numerical examples are presented to demonstrate the robustness, expected accuracy, and high resolution of the proposed method.Comment: 56 pages, 18 figure

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.Comment: 47 pages, 14 figure