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

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.