Numerical methods for wave equations need dissipation to remain stable. Without it, errors accumulate and solutions blow up. The standard approach adds artificial viscosity — an explicit damping term chosen by the practitioner, tuned to balance accuracy against stability.
Treating time as just another spatial dimension in a summation-by-parts framework, using a dual (non-diagonal) pairing between trial and test spaces, dissipation emerges automatically from the geometric structure of the discretization. No artificial term is added. The dissipation is a consequence of the asymmetric pairing.
The same framework simultaneously produces adjoint-consistent discretizations for inverse problems. The numerical adjoint exactly matches the continuous adjoint — a property that usually requires careful separate construction. Here it falls out of the same geometric structure that produces the temporal dissipation.
The reason this works is that the dual pairing introduces a controlled asymmetry between how information flows forward in time (the wave equation) and backward (the adjoint). In the continuous problem, this asymmetry is trivial — the adjoint of a wave operator is just the wave operator with reversed time. But in the discrete problem, preserving this relationship while also maintaining stability is typically a separate engineering problem. The dual SBP framework makes both consequences of a single structural choice.
The broader implication is about where properties come from. Dissipation in numerical methods has traditionally been viewed as something you impose — a correction to the physics to make the numerics work. Here, it's something the geometry gives you. The difference matters: imposed dissipation requires tuning and can interfere with accuracy. Structural dissipation is automatically calibrated to what the discretization needs.
Structure gives you for free what ad hoc methods struggle to add.