High-Order Fully Discrete Entropy-Stable DG Methods with Compact Explicit Single-Step Time Discretizations

1. Introduction.

• The approach responds to the broader challenge that semidiscrete entropy stability does not automatically persist under explicit high-order time integration.Existing fully discrete alternatives include implicit methods and relaxation Runge–Kutta schemes, with local entropy control in the latter requiring cellwise root-finding.Source: §1 · pp. 1–2
• The framework builds on prior cell-average-preserving scaling, which uses a weak entropy inequality for updated cell averages as the feasibility condition for terminal polynomial scaling.That algebraic limiter requires no nonlinear solver, accommodates finite families of convex entropy pairs, and yields a Lax–Wendroff-type theorem [17, 18].Source: §1 · p. 2
• A single end-of-step, cell-average-preserving scaling is sought to enforce a two-level strong cell entropy inequality without sacrificing high-order temporal accuracy.Earlier SSP multistep bounds involve entropies from multiple past levels, while limiting every SSP-RK stage can degrade spatiotemporal accuracy to first order [17] [29].Source: §1 · p. 2
• Standard SSP-RK3 DG updates can violate the necessary cell-average entropy condition at arbitrarily small positive CFL numbers, preventing any terminal cell-average-preserving correction from ensuring global entropy nonincrease.The counterexamples trace the overshoot to successive stages propagating interface slopes across multiple cells within one step.Source: §1 · p. 2
• The framework derives feasible numerical entropy-flux intervals for compact single-step DG methods and uses a projected target flux to secure the weak cell entropy inequality needed for terminal scaling.The intervals are proved nonempty under a CFL restriction; the approach covers cRKDG, LWDG, and ADER-DG.Source: §1 · pp. 2–3

2.1. Conservation laws and entropy inequalities.

• Admissible weak solutions satisfy an entropy inequality, which implies nonincreasing total entropy on a periodic domain.Smooth solutions satisfy the corresponding entropy conservation law.Source: §2.1 · pp. 3–4

2.2. DG approximations and fully discrete entropy stability.

• A terminal scaling correction exists if and only if the updated cell average satisfies the weak cell entropy inequality; summing that inequality gives a necessary global condition on a periodic mesh.The cell-average update is fixed by conservation, so violating the global condition rules out any choice of shared entropy fluxes and scaling factors.Source: §2.2 · p. 5

2.3. A counterexample for the standard SSP-RK3 DG method.

• In the SSP-RK3 counterexample, successive stages expand the spatial stencil, propagating slopes from an upward jump into the second cell to its right.The construction uses Burgers’ equation with an exponential entropy pair and piecewise-constant initial data.Source: §2.3 · p. 6
• At arbitrarily small positive CFL numbers, an SSP-RK3 DG update can increase total cell-average entropy, ruling out any terminal cell-average-preserving entropy correction.In the counterexample, one cell’s entropy after the step exceeds the initial entropy of all twelve potentially affected cells.Source: §2.3 · p. 7

2.4. Compact RKDG and the preceding counterexample.

• For the piecewise-constant counterexample data, compact RKDG reduces to a forward Euler update whose affected cell-average pairs preserve the necessary global entropy condition.Only four cells adjacent to the two jumps change their averages.Source: §2.4 · pp. 7–8

• The framework addresses a general class of compact explicit single-step DG methods.Source: §3 · p. 8
• The section establishes cellwise entropy stability for the class of methods under study.Source: §3 · p. 8
• The section also establishes local accuracy preservation for these methods.Source: §3 · p. 8

3.1. Compact explicit single-step DG methods.

• Compact single-step updates use cell-local volume fluxes and interface fluxes depending only on the two adjacent cells.On piecewise-constant data, the interface flux reduces to the physical flux.Source: §3.1 · p. 8
• The analysis assumes a uniformly convex entropy on the state range under consideration.The stated condition is 0 < µ ≤ U′′(u) ≤ L.Source: §3.1 · p. 9
• The framework requires strict entropy dissipation, uniform Lipschitz continuity, and reduction to the physical flux on constant data.These are properties (P1)–(P3) used in the analysis.Source: §3.1 · p. 9
• Standard SSP-RKDG schemes generally fail the two-cell Lipschitz and constant-reduction properties because internal stages couple neighboring cells.For the LF flux, the strict dissipation property holds when its parameter strictly bounds the wave speed.Source: §3.1 · p. 9

3.2. A reformulation of a first-order scheme.

• The weak cell entropy inequality for a first-order update can be ensured by requiring half-cell entropy inequalities at each interface.Convexity combines the two half-cell inequalities into the cell inequality.Source: §3.2 · p. 10
• A shared numerical entropy flux must lie in the interval defined by the half-cell constraints to ensure the cell entropy inequality.This gives an explicit local set of sufficient flux choices for the fixed conservative update.Source: §3.2 · p. 10
• For the LF flux (2.5), the proper LF entropy flux (3.6) belongs to the interval [ˆFlo, ˆFhi], which is nonempty under a suitable CFL condition.Source: §3.2 · pp. 10–11
• For high-order compact updates, the Jensen residual from polynomial fluctuations supplies entropy budget to absorb the conservative-flux perturbation.The budget helps keep the perturbed feasible intervals nonempty as intra-cell fluctuations are restored.Source: §3.2 · p. 11

3.3. High-order feasible intervals.

• The feasible intervals are nonempty under a CFL restriction for the compact single-step framework.This permits selecting a shared numerical entropy flux that satisfies the weak cell entropy inequality.Source: §3.3 · pp. 11–12
• Taking the minimum of the individual scaling factors enforces strong cell entropy inequalities for a finite family of convex entropy pairs simultaneously.The framework also states a Lax–Wendroff-type result for strong L1 limits satisfying the enforced entropy inequalities.Source: §3.3 · pp. 12–13
• Projecting a high-order target entropy flux onto the feasible interval supplies the flux needed for terminal polynomial scaling.The resulting weak cell entropy inequality is the prerequisite for the strong cell entropy inequality after scaling.Source: §3.3 · pp. 12–13

3.4. High-order entropy-flux design and accuracy preservation.

• The local variation scale is O(h) away from critical points and O(h^2) within O(h) of a nondegenerate extremum.This scaling is part of the accuracy analysis for fixed k ≥ 2.Source: §3.4 · pp. 13–14
• In smooth regions under the stated assumptions, the high-order target flux lies inside the feasible interval, so the projection is inactive.The accuracy result applies for sufficiently small h and CFL parameter λ.Source: §3.4 · p. 14
• The selected flux preserves high-order accuracy uniformly across smooth patches, including cells near nondegenerate extrema.The result is established under smooth-solution and approximation-accuracy assumptions.Source: §3.4 · pp. 13–14
• A unified target construction uses local predictor traces and a Lax–Friedrichs entropy flux for schemes based on the LF conservative flux.The target satisfies the stated nonlinear consistency assumption when the predictor traces meet the proposition’s conditions.Source: §3.4 · pp. 15–16
• The construction is presented for cRKDG, LWDG, and ADER-DG while leaving their preliminary DG solution updates unchanged.The target entropy flux or predictor is the focus of the construction.Source: §3.4 · p. 16

3.5.1. Compact RKDG.

• The cRKDG construction uses fractional-step local RK updates or a continuous extension to build a temporal predictor satisfying the required accuracy conditions.The continuous extension reuses full-step local stage derivatives; the described option uses only the first two derivatives and no extra operator evaluations.Source: §3.5.1 · pp. 16–17
• A stage-averaged entropy flux can reduce cRKDG single-step accuracy to O(h^2) near smooth extrema for fixed λ, whereas the unified target gives D_i = 0 and θ_i = 1.The cited cRKDG example uses a P2 update with Heun’s third-order RK method; the reported numerical tests did not show order reduction, possibly because their CFL number was 0.01.Source: §3.5.1 · p. 17

3.5.2. Lax–Wendroff

• LWDG uses the Cauchy–Kowalevski relation to replace temporal Taylor coefficients with spatial derivatives and satisfies the predictor and interface-flux accuracy requirements.Its construction uses a Taylor polynomial, a volume flux, and a Lax–Friedrichs interface flux.Source: §3.5.2 · p. 17

3.5.3. ADER-DG.

• ADER-DG builds a cell-local space–time predictor and uses the same positive time quadrature for its conservative and target entropy fluxes.With quadrature exact through degree k + 2, the conservative-flux matching term vanishes, yielding the stated target-flux accuracy condition.Source: §3.5.3 · p. 18

3.6. Extensions to systems and multidimensional problems.

• For systems, the entropy fluxes and feasible intervals remain scalar while conservative fluxes and virtual states are vector-valued.The nonempty-interval result assumes entropy-evaluation states lie in a convex admissible range where the entropy Hessian bounds hold.Source: §3.6 · p. 18
• On d-dimensional Cartesian meshes, equal-weight directional virtual states yield facewise feasible entropy-flux intervals under a dimension-dependent CFL condition.A single scalar scaling factor θ_K is applied to all state components, preserving the vector cell average and enforcing the strong cell entropy inequality.Source: §3.6 · p. 18

4.1. Scalar problems.

• All three methods—cRKDG, LWDG, and ADER-DG—show optimal convergence rates for linear advection with exponential entropy.The comparison uses L2 errors at T = 1; Figure 2 displays the errors and reference orders.Source: §4.1 · p. 19, Figure 2 · p. 19
• Five shifted arctan entropies substantially reduce discontinuity overshoots versus one entropy, while nine shifts further damp oscillations at the cost of a slightly broader transition.The profile comparison is at T = 2π on N = 160 cells.Source: §4.1 · p. 19, Figure 3 · p. 20
• For smooth Burgers flow, cRKDG, LWDG, and ADER-DG all achieve the expected convergence orders with exponential entropy.The L2 error test is at T = 0.6, before the solution’s stated smoothness time T_s = 1.Source: §4.1 · p. 19, Figure 4 · p. 20
• For both Buckley–Leverett Riemann problems, simultaneous enforcement of U(1) and U(2) satisfies both cell entropy inequalities to machine precision and captures the admissible composite waves.Enforcing either entropy alone produces a nonphysical plateau or large oscillations, and leaves the unenforced entropy with positive cell residuals.Source: §4.1 · p. 20, Figure 5 · p. 21, Table 1 · p. 21

4.2. One-dimensional Euler equations.

• The smooth isentropic-wave Euler test produces comparable density L2 errors and convergence rates for the entropy-stabilized and baseline schemes.The comparison is at T = 0.2.Source: §4.2 · pp. 21–22, Table 2 · p. 22
• In the Shu–Osher test, the entropy-stabilized density profile resolves the shock sharply and captures post-shock density waves against a 2000-cell reference.The displayed solution is at T = 1.8 on N = 200 cells, with the lower panel enlarging post-shock oscillations.Source: §4.2 · p. 22, Figure 6 · p. 22

4.3. Two-dimensional Euler equations.

• For the isentropic vortex, entropy-stabilized and baseline schemes achieve comparable third-order convergence with nearly identical fine-grid errors.The density comparison is at T = 10 on N × N meshes.Source: §4.3 · p. 23, Table 3 · p. 23
• The Double Mach reflection computation captures Mach stems and Kelvin–Helmholtz roll-up along the primary slip line.The density is shown at T = 0.2 on 960 × 240 cells with a positivity-preserving limiter.Source: §4.3 · p. 23, Figure 7 · p. 23

5. Concluding remarks.

• Scaling the preliminary DG polynomial toward its cell average once per step enforces a strong cell entropy inequality while preserving conservation.Standard SSP-RK3 DG updates can violate the necessary cell-average entropy condition at arbitrarily small CFL numbers, while compact single-step methods admit feasible intervals under a CFL condition.Source: §5 · pp. 23–24
• Projecting a compatible high-order target entropy flux onto the feasible interval guarantees the weak cell entropy condition and preserves optimal single-step accuracy, including near nondegenerate extrema.The unified Lax–Friedrichs target construction applies to cRKDG, LWDG, and ADER-DG.Source: §5 · p. 24
• The framework supports multiple simultaneous entropy constraints and a Lax–Wendroff-type result for strong L1 limits, while experiments report optimal convergence, multi-entropy control, and robustness.Extensions to unstructured meshes and more complex systems are identified as future work.Source: §5 · p. 24