Entropy Correction Artificial Viscosity

• Entropy correction artificial viscosity (ECAV) is defined as a method that computes a local, parameter-free viscosity coefficient based on the cell's entropy deficit to enforce a discrete entropy inequality.
• ECAV enhances high-order DG and finite-difference methods by providing minimal, targeted dissipation that preserves contact discontinuities and robustly handles shocks.
• The approach distinguishes itself by distributing dissipation among multiple viscosity models, avoiding parameter tuning typical in traditional shock-capturing techniques.

Searching arXiv for recent and foundational papers on entropy correction artificial viscosity. Entropy correction artificial viscosity (ECAV) is an approach for enforcing a semi-discrete entropy inequality through an entropy dissipative correction term. In high-order discretizations of nonlinear conservation laws, it replaces or supplements more traditional entropy-stable constructions based on algebraic flux differencing, summation-by-parts (SBP) operators, and entropy conservative two-point fluxes. In the ECAV formulation developed for discontinuous Galerkin (DG) methods, the artificial viscosity coefficient is determined from the local violation of a cell entropy inequality and the local entropy dissipation, yielding a parameter-free, cell-local correction that is zero when the entropy inequality is already satisfied (Chan, 27 Jan 2025). Subsequent work extended the construction to multiple artificial viscosities, to finite difference discretizations, and to analyses of the viscous DG discretization itself, including contact preservation and an $O(h)$ upper bound on the ECAV coefficient (Park et al., 3 Apr 2026, Christner et al., 28 Aug 2025, Fleet et al., 26 Feb 2026).

1. Conceptual basis and development

Entropy stable methods have become increasingly popular in the field of computational fluid dynamics because schemes which satisfy some form of a discrete entropy inequality typically behave much more robustly, and do so in a way that is hyperparameter free (Christner et al., 28 Aug 2025). In DG, one established route to entropy stability uses entropy conservative fluxes in an algebraic flux differencing formulation together with SBP discretization matrices. However, explicit expressions for such two-point finite volume fluxes may not be available for all systems, or may be computationally expensive to compute (Chan, 27 Jan 2025). ECAV was proposed as an alternative approach to constructing entropy stable DG methods using an entropy correction artificial viscosity, where the artificial viscosity coefficient is determined based on the local violation of a cell entropy inequality and the local entropy dissipation (Chan, 27 Jan 2025).

The immediate precursor of this line of work is the entropy correction of Abgrall, Offner, and Ranocha (2022), which was reinterpreted and modified into an artificial-viscosity form. The resulting method recovers the same global semi-discrete entropy inequality that is satisfied by entropy stable flux differencing DG methods, while avoiding dependence on bespoke entropy conservative flux functions (Chan, 27 Jan 2025). This design choice situates ECAV between classical shock-capturing artificial viscosity methods, which typically use sensors and tunable parameters, and fully entropy-conservative or entropy-stable flux differencing schemes, which derive robustness from discrete thermodynamic structure.

A central organizing principle is that the entropy inequality is enforced by adding only as much dissipation as needed. This minimality is explicit in the coefficient formulas used in both DG and finite difference settings. The method is therefore not a generic viscosity model; it is an entropy correction whose magnitude is computed from an algebraic entropy deficit and an associated dissipative mechanism (Chan, 27 Jan 2025, Christner et al., 28 Aug 2025).

2. Mathematical formulation in high-order DG

The target problem is a system of conservation laws

$$\frac{\partial \boldsymbol{u}}{\partial t} + \sum_{i = 1}^d \frac{\partial \boldsymbol{f}_i(\boldsymbol{u})}{\partial x_i} = \boldsymbol{0},$$

equipped with a convex entropy $S(u)$, entropy variables $v(u) = \frac{\partial S}{\partial u}$, and entropy fluxes $F_m(u)$. The entropy inequality reads

$$\frac{\partial S(u)}{\partial t} + \sum_{m = 1}^d \frac{\partial F_m(u)}{\partial x_m} \leq 0$$

or, over a DG cell $D$,

$$\int_D \frac{\partial S(u)}{\partial t} + \int_{\partial D} \sum_{m=1}^d \left( v^T f_m(u) - \psi_m(u) \right) n_m \le 0,$$

where $\psi_m(u)$ is the entropy potential (Park et al., 3 Apr 2026).

In the monolithic DG construction, artificial viscosity is introduced through a modified PDE of the form

$$\frac{\partial \boldsymbol{u}}{\partial t} + \sum_{i=1}^d \frac{\partial \bm{f}_i(\boldsymbol{u})}{\partial x_i} = \sum_{i,j=1}^d \frac{\partial}{\partial x_i} \left( \epsilon_k(\boldsymbol{u}) \bm{K}_{ij} \frac{\partial \bm{v}}{\partial x_j} \right),$$

with $$\frac{\partial \boldsymbol{u}}{\partial t} + \sum_{i = 1}^d \frac{\partial \boldsymbol{f}_i(\boldsymbol{u})}{\partial x_i} = \boldsymbol{0},$$0 a per-element, non-negative artificial viscosity coefficient and $$\frac{\partial \boldsymbol{u}}{\partial t} + \sum_{i = 1}^d \frac{\partial \boldsymbol{f}_i(\boldsymbol{u})}{\partial x_i} = \boldsymbol{0},$$1 positive semidefinite. A common choice is Laplacian viscosity with $$\frac{\partial \boldsymbol{u}}{\partial t} + \sum_{i = 1}^d \frac{\partial \boldsymbol{f}_i(\boldsymbol{u})}{\partial x_i} = \boldsymbol{0},$$2 (Park et al., 3 Apr 2026).

The DG discretization uses a BR1 treatment of the viscous term. Auxiliary variables $$\frac{\partial \boldsymbol{u}}{\partial t} + \sum_{i = 1}^d \frac{\partial \boldsymbol{f}_i(\boldsymbol{u})}{\partial x_i} = \boldsymbol{0},$$3 approximate entropy-variable gradients, viscous fluxes $$\frac{\partial \boldsymbol{u}}{\partial t} + \sum_{i = 1}^d \frac{\partial \boldsymbol{f}_i(\boldsymbol{u})}{\partial x_i} = \boldsymbol{0},$$4 are formed from $$\frac{\partial \boldsymbol{u}}{\partial t} + \sum_{i = 1}^d \frac{\partial \boldsymbol{f}_i(\boldsymbol{u})}{\partial x_i} = \boldsymbol{0},$$5, and the viscous contribution $$\frac{\partial \boldsymbol{u}}{\partial t} + \sum_{i = 1}^d \frac{\partial \boldsymbol{f}_i(\boldsymbol{u})}{\partial x_i} = \boldsymbol{0},$$6 is inserted into the weak form (Park et al., 3 Apr 2026). The entropy correction is driven by a local, cell-based entropy residual

$$\frac{\partial \boldsymbol{u}}{\partial t} + \sum_{i = 1}^d \frac{\partial \boldsymbol{f}_i(\boldsymbol{u})}{\partial x_i} = \boldsymbol{0},$$7

In the related DG construction with entropy projection, the residual is written

$$\frac{\partial \boldsymbol{u}}{\partial t} + \sum_{i = 1}^d \frac{\partial \boldsymbol{f}_i(\boldsymbol{u})}{\partial x_i} = \boldsymbol{0},$$8

where $$\frac{\partial \boldsymbol{u}}{\partial t} + \sum_{i = 1}^d \frac{\partial \boldsymbol{f}_i(\boldsymbol{u})}{\partial x_i} = \boldsymbol{0},$$9 is the $S(u)$0 projection onto degree-$S(u)$1 polynomials and $S(u)$2 is the entropy-projected conservative variable (Chan, 27 Jan 2025).

The minimal elementwise coefficient is then chosen as

$S(u)$3

with a small regularization of the denominator for numerical stability (Park et al., 3 Apr 2026). In the DG entropy-projection formulation, the same construction appears as

$S(u)$4

and if $S(u)$5, no viscosity is applied (Chan, 27 Jan 2025).

The entropy dissipation guarantee is explicit: for the BR1 discretization,

$S(u)$6

Thus, artificial viscosity applied in this way strictly enforces the discrete entropy inequality (Park et al., 3 Apr 2026).

3. Viscous discretization, size of the coefficient, and contact preservation

A distinct line of analysis concerns the choice of viscous DG discretization for ECAV. When the artificial viscosity is discretized using a local discontinuous Galerkin (LDG) method, an $S(u)$7 upper bound on the ECAV coefficient can be proved (Fleet et al., 26 Feb 2026). This upper bound indicates that ECAV does not result in a restrictive time-step condition, so the explicit time-step remains hyperbolic rather than parabolic in scale (Fleet et al., 26 Feb 2026).

The LDG choice is motivated by structural properties of the discrete gradient operator. In the analysis summarized for (Fleet et al., 26 Feb 2026), LDG yields a local gradient operator whose norm is bounded below by the true gradient norm, and the nullspace of the LDG gradient for polynomials is only the constant function. By contrast, BR1 can admit non-constant null modes, which can lead to numerical pathologies and spuriously large viscosity coefficients. A plausible implication is that the ECAV mechanism depends not only on the entropy residual, but also on whether the viscous discretization provides a robust denominator in the coefficient formula.

The same analysis shows that ECAV is contact preserving (Fleet et al., 26 Feb 2026). With piecewise constant solutions such as stationary contacts, the entropy residual vanishes, so ECAV adds no artificial viscosity. If the underlying DG scheme is contact preserving, ECAV with LDG does not alter that property (Fleet et al., 26 Feb 2026). This distinguishes ECAV from many traditional shock-capturing artificial viscosity methods, whose dissipation is selected through problem-dependent indicators and thresholds and can smear contacts.

The contrast with heuristic shock capturing is consequential. Traditional approaches add viscosity based on sensors or smoothness indicators and typically require parameter tuning. ECAV computes its coefficient directly from the algebraic balance of entropy at the element level and injects only as much dissipation as needed to restore the inequality (Fleet et al., 26 Feb 2026). In this sense, the artificial viscosity is neither prescribed a priori nor globally calibrated; it is an entropy deficit correction.

4. Multiple viscosities and targeted dissipation

The original DG formulation used a single monolithic viscosity coefficient. This was generalized to multiple dissipation models, including viscosity and thermal diffusivity, by writing

$S(u)$8

with each viscous term $S(u)$9 assigned its own elementwise coefficient $v(u) = \frac{\partial S}{\partial u}$0 (Park et al., 3 Apr 2026). The coefficients are allocated by solving an elementwise constrained minimization problem,

$v(u) = \frac{\partial S}{\partial u}$1

where $v(u) = \frac{\partial S}{\partial u}$2 and $v(u) = \frac{\partial S}{\partial u}$3 denotes the entropy-dissipative contribution of model $v(u) = \frac{\partial S}{\partial u}$4 (Park et al., 3 Apr 2026). With equal weighting, the closed-form solution is

$v(u) = \frac{\partial S}{\partial u}$5

This construction divides dissipation among models proportional to their relative contribution to entropy dissipation, for minimal overall added dissipation (Park et al., 3 Apr 2026). The purpose is not merely algebraic convenience. Different artificial viscosity mechanisms can be directed toward different numerical or physical artifacts: Laplacian viscosity for general regularization, thermal diffusivity for false heating or overheating, and spectral vanishing viscosity (SVV) for under-resolved turbulence or shear (Park et al., 3 Apr 2026).

Model	Role described in the literature	ECAV allocation principle
Laplacian viscosity	General conservative regularization	Included in the constrained minimization
Thermal viscosity	Acts on energy; reduces spurious temperature spikes	Activated through its entropy-dissipative contribution
SVV	Targets high modes and under-resolved turbulence/shear	Receives dissipation only where needed

The reported examples illustrate the selectivity of the multiple-viscosity formulation. In a 1D receding flow, adding thermal artificial viscosity alongside Laplacian artificial viscosity reduces spurious temperature spikes while leaving density and pressure profiles largely unaffected (Park et al., 3 Apr 2026). In a 2D Kelvin–Helmholtz problem, the SVV coefficient is only activated where needed, in regions of high shear or turbulence, while in a 2D Riemann problem the extra SVV coefficient is essentially zero when the additional mechanism is physically unnecessary (Park et al., 3 Apr 2026). These results support the characterization of ECAV as minimally dissipative and physically targeted, rather than monolithic.

5. Finite-difference ECAV and the relation to knapsack limiting

ECAV was subsequently introduced for finite difference discretizations in an SBP flux-differencing framework (Christner et al., 28 Aug 2025). In that setting, the high-order residual at node $v(u) = \frac{\partial S}{\partial u}$6 is written

$v(u) = \frac{\partial S}{\partial u}$7

where $v(u) = \frac{\partial S}{\partial u}$8 is the high-order, central, high-accuracy flux, $v(u) = \frac{\partial S}{\partial u}$9 is a symmetric non-negative artificial viscosity coefficient, and $F_m(u)$0 are geometric weights (Christner et al., 28 Aug 2025). The cell entropy inequality becomes a nodal constraint,

$F_m(u)$1

which reduces to a linear inequality $F_m(u)$2 with

$F_m(u)$3

The minimal local correction solves

$F_m(u)$4

and has the explicit solution

$F_m(u)$5

Conservation is recovered by symmetrization, $F_m(u)$6 (Christner et al., 28 Aug 2025).

This finite-difference version preserves high order accuracy in sufficiently smooth conditions, is entropy stable, and is hyperparameter free (Christner et al., 28 Aug 2025). It also clarifies the relation between ECAV and knapsack limiting (KL). KL blends a high-order scheme with a low-order, positivity preserving, and entropy stable scheme, whereas ECAV adds the minimal local artificial viscosity needed to satisfy the cell entropy inequality (Christner et al., 28 Aug 2025). KL can additionally incorporate positivity constraints for the compressible Euler and Navier–Stokes equations; ECAV, by itself, does not preserve positivity (Christner et al., 28 Aug 2025).

The distinction is structural. ECAV is a viscosity-based entropy correction; KL is a constrained flux blending strategy. Yet, for the special case where the low-order flux is local Lax–Friedrichs, the KL flux can be written as a special case of the ECAV form with $F_m(u)$7 scaled by the maximum wavespeed (Christner et al., 28 Aug 2025). This places both methods within a common framework of local convex optimization driven by entropy admissibility.

6. Related interpretations, neighboring methods, and recurring misconceptions

ECAV sits within a broader landscape of entropy-based artificial dissipation, but it is not identical to older entropy viscosity methods. In entropy viscosity, the local viscosity coefficient is typically

$F_m(u)$8

with $F_m(u)$9 and a scaling exponent $$\frac{\partial S(u)}{\partial t} + \sum_{m = 1}^d \frac{\partial F_m(u)}{\partial x_m} \leq 0$$0 that strongly affects performance (Kornelus et al., 2017). The analysis of shock problems in (Kornelus et al., 2017) shows that $$\frac{\partial S(u)}{\partial t} + \sum_{m = 1}^d \frac{\partial F_m(u)}{\partial x_m} \leq 0$$1 is preferable to $$\frac{\partial S(u)}{\partial t} + \sum_{m = 1}^d \frac{\partial F_m(u)}{\partial x_m} \leq 0$$2 for robust shock capturing in high-order methods, especially with moving shocks and multiple shocks of different strengths. ECAV differs in that its coefficient is not set by a residual scaling law with tunable parameters; it is determined by the amount of entropy dissipation required to close a discrete entropy imbalance (Chan, 27 Jan 2025, Christner et al., 28 Aug 2025).

Another neighboring line concerns entropy correction through filters. In DG discretizations stabilized via SIAC filters, entropy correction is written in a filtered form such as

$$\frac{\partial S(u)}{\partial t} + \sum_{m = 1}^d \frac{\partial F_m(u)}{\partial x_m} \leq 0$$3

with conservative filter correction and optional dissipative augmentation through an artificial viscosity estimate (Picklo et al., 2023). This is an entropy-correction strategy rather than the elementwise coefficient construction that defines ECAV. It nevertheless illustrates a shared theme: entropy correction can be appended to a high-order method while preserving conservation and controlling dissipation (Picklo et al., 2023).

A further misconception concerns the sign of artificial viscosity. Artificial viscosity has traditionally been interpreted as a positive, spatially acting regularization, and local entropy stability is often linked to pointwise positivity. The data-driven study of linear convection in (Neelan, 8 Feb 2026), however, reports locally negative artificial viscosity near extrema while the numerical solution remains stable and nearly exact. The paper resolves this by reinterpreting artificial viscosity as a space-time closure that compensates unresolved truncation errors while enforcing entropy stability through global dissipation balance rather than pointwise positivity. Within that framework, entropy stability constrains the integrated dissipation budget rather than the pointwise sign of spatial viscosity (Neelan, 8 Feb 2026). This does not redefine ECAV directly, but it broadens the conceptual context in which entropy-based viscosity is interpreted.

Finally, high-order artificial dissipation in SBP-compatible correction procedure via reconstruction (CPR) schemes provides another adjacent perspective. There, stable artificial dissipation operators are built in the SBP form

$$\frac{\partial S(u)}{\partial t} + \sum_{m = 1}^d \frac{\partial F_m(u)}{\partial x_m} \leq 0$$4

rather than by naïve discretization, in order to guarantee conservation and discrete $$\frac{\partial S(u)}{\partial t} + \sum_{m = 1}^d \frac{\partial F_m(u)}{\partial x_m} \leq 0$$5 stability (Ranocha et al., 2016). This reinforces a point that also appears in ECAV: the entropy or energy effect of artificial viscosity depends as much on the discretization of the viscous term as on the scalar coefficient itself.

Across these strands, the unifying theme is that entropy correction artificial viscosity is best understood as a minimally dissipative, locally computed mechanism for enforcing a discrete entropy inequality. Its later developments show that the choice of viscous discretization, the possibility of multiple targeted viscosities, and the extension to finite differences are not peripheral refinements; they are integral to how entropy correction is made accurate, contact preserving, and compatible with high-order explicit solvers (Fleet et al., 26 Feb 2026, Park et al., 3 Apr 2026, Christner et al., 28 Aug 2025).