Data-Driven Discovery of Sign-Indefinite Artificial Viscosity for Linear Convection -- A Space-Time Reconvolution Perspective

TL;DR

The paper addresses stabilizing hyperbolic discretizations by reinterpreting artificial viscosity as a space–time closure rather than a purely spatial diffusion. Using gradient-based optimization with automatic differentiation on an unstable FTCS discretization of $u_t + a u_x = 0$, it learns a viscosity field that locally becomes sign-indefinite yet yields stable, near-exact solutions; this motivates viewing viscosity as a reconvolution operator compensating truncation errors. A space–time closure framework with coefficients $μ_s(x,t)$ and $μ_t(x,t)$ is developed, and the Lax–Wendroff scheme is recast as a degenerate time-to-space projection, explaining how global entropy stability constrains the integrated dissipation rather than the pointwise sign. The results unify data-driven closures with Tadmor-type entropy-stable central schemes, spectral viscosity, and compact flux corrections, offering a principled approach to adaptive, interpretable stabilizations for both linear and nonlinear conservation laws.

Abstract

Artificial viscosity is traditionally interpreted as a positive, spatially acting regularization introduced to stabilize numerical discretizations of hyperbolic conservation laws. In this work, we report a data-driven discovery that motivates a reinterpretation of this classical view. We consider the linear convection equation discretized using an unstable FTCS scheme augmented with a learnable artificial viscosity. Using automatic differentiation and gradient-based optimization, the viscosity field is inferred by minimizing the error with respect to the exact solution, without imposing any sign constraints. The optimized viscosity consistently becomes locally negative near extrema, while the numerical solution remains stable and nearly exact. This behavior is not readily explained within classical modified equation analysis and Lax-Wendroff-type arguments, which predict a strictly positive effective viscosity. To resolve this apparent contradiction, we reinterpret 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 this framework, the Lax-Wendroff scheme corresponds to a degenerate projection in which temporal truncation errors are eliminated and reintroduced as spatial diffusion. We show that entropy stability constrains the integrated dissipation budget rather than the pointwise sign of spatial viscosity. As a result, locally negative viscosity naturally emerges as a numerical reconvolution operator that compensates for dispersive truncation errors. Negative viscosity is therefore not an unphysical diffusion process, but a scheme- and grid-dependent correction mechanism.

Figures (2)

• Figure 1: Comparison of learned artificial viscosity
• Figure 2: Comparison of learned artificial viscosity