Dice Question Streamline Icon: https://streamlinehq.com

Alternative discretization schemes and resulting wave structures

Develop and analyze alternative spatial discretizations (beyond the standard central difference) for the discrete conservation law associated with u_t + [\Phi(u)]_x = 0, derive the corresponding lattice equations, and determine the resulting nonlinear dispersive wave structures produced by Riemann initial data.

Information Square Streamline Icon: https://streamlinehq.com

Background

The studied lattice du_n/dt + u_n{\widetilde{p}}(u_{n+1} − u_{n-1}) = 0 uses a central difference representation of the spatial derivative, yielding DSWs and RWs under Riemann-type initial data.

The authors suggest exploring other discretization schemes for the scalar conservation law u_t + [\Phi(u)]_x = 0 to generate distinct lattices that might exhibit richer or different dispersive wave phenomena.

References

There are still a variety of interesting open questions remaining, and we only list a few of them. Lastly, it may also be relevant to consider some other discretization scheme for the scaler conservation law in Eq.~eq: general discrete conservation law. We notice that our lattice eq: extension 2 uses the standard central difference formula to discretize the spatial derivative of $u_x$, so with a different numerical scheme, one shall obtain a distinct lattice, and it may have more abundant structures of wave.

eq: general discrete conservation law:

ut+[Φ(u)]x=0,u_t + \left[\Phi(u)\right]_x = 0,

eq: extension 2:

dundt+(un)p~(un+1−un−1)=0.\frac{d u_{n}}{dt} + \left(u_n\right)^{\widetilde{p}}\left(u_{n+1} - u_{n-1}\right) = 0.

— Quasi-continuum approximations for nonlinear dispersive waves in general discrete conservation laws (2509.04630 - Yang, 4 Sep 2025) in Conclusions and future directions