Zakerzadeh and May (2016) proved that a space-time discontinuous Galerkin (DG) method written in entropy variables and stabilized solely by a nonlinear residual-based shock-capturing viscosity converges to an entropy measure-valued (emv) solution of general nonlinear hyperbolic systems of conservation laws in several space dimensions, for arbitrary polynomial order. The theorem is, however, conditional on assumed a priori bounds: a uniform L∞L_\infty bound on the approximate entropy variables, uniform spectral bounds on the entropy symmetrizer and the shock-capturing diffusion matrix, convex space-time elements, and a shock-capturing operator that is not dimensionally consistent. The open problem addressed here is to complete this convergence theory: establish the required boundedness from first principles or remove it, construct and analyze a dimensionally consistent shock-capturing operator without streamline diffusion, extend the analysis beyond convex meshes, and complement the theory with a rigorous statistical (ensemble-based) numerical verification of emv convergence for this scheme class. Later literature built surrounding frameworks (approximate emv solutions, statistical solutions, compensated compactness, dissipative weak solutions) but left each of these items open.

Background

Nonlinear hyperbolic systems of conservation laws ut+∑k=1d∂xkfk(u)=0u_t+\sum_{k=1}^d \partial_{x_k}f^k(u)=0, where uu is a vector of conserved quantities and each flux fkf^k is nonlinear, model gas dynamics, shallow water flow, and magnetohydrodynamics. Their solutions develop discontinuities (shocks) in finite time even from smooth initial data, so weak solutions must be sought, and an entropy pair — a strictly convex entropy function UU with compatible entropy fluxes FkF^k — is used to single out physically admissible ones through the inequality ∂tU(u)+∑k∂xkFk(u)≤0\partial_t U(u)+\sum_k \partial_{x_k}F^k(u)\le 0. Introducing entropy variables v=U′(u)v=U'(u) symmetrizes the system (the flux Jacobians become symmetric in these variables); the Jacobian uv=∂u/∂vu_v=\partial u/\partial v is called the symmetrizer. This transformation underlies a successful design principle for high-order numerical methods: discretize the equations in entropy variables using an entropy-stable numerical flux (one whose interface jumps dissipate a discrete entropy), which yields a discrete entropy inequality.

For high-order discretizations the remaining obstacle is spurious oscillations near shocks. Space-time discontinuous Galerkin (DG) methods — in which the solution is a piecewise polynomial on space-time elements and continuity is enforced only weakly — can be stabilized by streamline diffusion (SD), a linear diffusion term acting along the space-time flow direction, and/or by a nonlinear shock-capturing (SC) artificial viscosity whose coefficient is proportional to a scaled residual of the PDE and hence is large only near discontinuities. Hiltebrand and Mishra (2014) proved that a space-time DG scheme in entropy variables with both SD and SC terms converges to an entropy measure-valued solution; Zakerzadeh and May (2016) removed the SD term and showed that the nonlinear SC viscosity alone suffices for entropy stability, entropy consistency, and convergence, for general systems and arbitrary order. A companion scalar convergence result for space-time DG (May and Zakerzadeh 2016) and a scalar residual-based-viscosity continuous-Galerkin precedent (Nazarov 2013) frame the same development.

The convergence target is the entropy measure-valued (emv) solution of DiPerna (1985): since approximate solutions need not converge strongly near shocks, one passes to a Young measure — a probability-measure-valued map νx,t\nu_{x,t} describing the one-point statistical distribution of limiting solution values — and requires that it satisfy the conservation law and the entropy inequality in an averaged sense. The compactness step in this program needs the family of approximations to be uniformly bounded in L∞L_\infty, and this is exactly where the theory becomes conditional. The convergence theorem of Zakerzadeh and May assumes: a uniform L∞L_\infty bound on the approximate entropy variables vhv^h; uniform spectral (eigenvalue) bounds on the symmetrizer uv(vh)u_v(v^h) and on the SC diffusion matrix D(vh)D(v^h); and convex space-time elements, which are needed for the approximation properties of the H1H^1 projection (the orthogonal projection of a function onto the discrete polynomial space, measured in the H1H^1 norm) used in the estimates. For scalar equations, monotone schemes satisfy a discrete maximum principle that delivers such bounds, but no such principle exists for systems or high-order schemes; the authors note that the spectral bounds appear attainable only for specific systems (shallow water, polytropic Euler) under additional physical constraints together with the L∞L_\infty bound. Two further acknowledged gaps remain: the proposed SC operator is not dimensionally consistent (its viscosity coefficient does not carry uniform scaling/units across space and time, unlike SD-based formulations of the Hiltebrand–Mishra line), and the numerical experiments, while supportive, do not verify emv convergence — the authors state that a statistical approach via ensemble generation of approximate Young measures would be required.

Problem Statement

Complete the convergence theory of entropy-variable shock-capturing space-time discontinuous Galerkin schemes — without streamline diffusion — for general nonlinear systems of conservation laws, by removing the ad hoc hypotheses of Zakerzadeh and May (2016). The common scientific objective is a convergence theory in which no boundedness property of the approximations is assumed without justification, expressed through four jointly supporting components: (1) Boundedness: either prove from first principles that the approximations satisfy a uniform L∞L_\infty bound on the entropy variables vhv^h together with the uniform spectral bounds on the symmetrizer uv(vh)u_v(v^h) and the shock-capturing diffusion matrix D(vh)D(v^h) — under explicitly stated, verifiable structural conditions on the system's entropy pair and on the data — or restructure the scheme and/or analysis so that convergence to an entropy measure-valued (emv) solution holds without these a priori bounds. (2) Dimensionally consistent shock-capturing operator: construct an SC operator whose viscosity coefficient carries consistent scaling (units) across space and time, and prove that it retains entropy stability, entropy consistency, and emv convergence without streamline diffusion, as the source paper explicitly left for future work. (3) Mesh generality: extend the convergence analysis beyond convex space-time elements, replacing or generalizing the H1H^1-projection approximation estimates for which convexity is currently required. (4) Statistical verification: provide a rigorous statistical numerical verification of emv convergence for this scheme class — an ensemble/Monte Carlo generation of approximate Young measures with quantified convergence of measured statistics — which the source paper calls for and which has so far been carried out only for entropy-stable finite-volume-type schemes. The target is that the convergence theorem for the SD-free SC-DG scheme for general systems no longer rests on unproven boundedness assumptions, on a dimensionally inconsistent operator, or on convexity-restricted meshes.

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