Great question—this is something a lot of folks get tripped up on when first working with TVD schemes in hydrodynamics, so let’s break this down clearly:

First, a critical clarification: TVD (Total Variation Diminishing) is a property of numerical discretization schemes, not the underlying partial differential equations (PDEs) themselves. The shallow water equations, kinematic wave equations, or any other flow model you’re looking at don’t have "TVD properties"—instead, we design numerical schemes to be TVD to prevent non-physical oscillations in the solution, especially near shocks or steep gradients.

Now, why don’t many researchers explicitly prove TVD for their work? There are a few key reasons:

• Existing mature theory for hyperbolic conservation laws: Shallow water and kinematic wave equations fall into the category of hyperbolic conservation laws, which have well-established numerical analysis frameworks. It’s a standard result that if you use a conservative, TVD-compliant discrete scheme (like finite difference/volume schemes with monotone fluxes, or popular variants like MUSCL-TVD, Roe-TVD, WENO-TVD) and adhere to the CFL stability condition, the numerical solution will be TVD. Most researchers assume readers are familiar with this foundational theory, so they don’t rehash the proof.
• Using pre-validated TVD schemes: If they’re applying a widely-used, already-proven TVD scheme (not inventing a new one), the TVD property of that scheme has already been demonstrated in classic numerical analysis literature. There’s no need to re-prove something that’s already a standard result in the field.

That said, there are cases where a researcher would need to explicitly verify TVD:

• If they’ve modified an existing TVD scheme (e.g., adding custom source term handling for complex bathymetry or friction in shallow water equations), the modification could break the original TVD conditions.
• If they’re developing an entirely new TVD scheme tailored to a specific variant of the flow equations, they’d need to prove that the new scheme maintains the TVD property.

As a side note: Kinematic wave equations are a simplified form of shallow water equations (ignoring inertial terms), so they still fit under the hyperbolic conservation law umbrella—meaning the same TVD scheme theory applies.

备注：内容来源于stack exchange，提问作者Panasun