A Semi‐Discrete Lagrangian‐Eulerian Numerical Scheme for Diffusive‐Dispersive Conservation Laws With Discontinuous Coefficient

In this work, we present an extension of the semi‐discrete Lagrangian‐Eulerian numerical scheme for diffusive‐dispersive conservation law problems, including a jump discontinuous flux function. As in the one‐dimensional scalar hyperbolic case, the no‐flow curves also organize the geometry of the method. On the computational side, we successfully reproduced one‐dimensional scalar problems from the literature, and to validate robustness under more extreme and challenging conditions, we extended the experiments to problems that go beyond what is currently available. In the two‐dimensional case, still in an exploratory manner, we obtained consistent results that indicate a natural extension of the method to multidimensional problems, guided by the no‐flow curves. On the theoretical side, with the coefficients of the diffusive and dispersive terms in balance, we showed the convergence of the numerical approximations to weak solutions and, in the hyperbolic limit problem, the convergence to the entropy solution. For convergence to the weak solution, we employed the Compensated Compactness Theorem, while for entropy convergence we used a Kružkov‐type criterion suitable for jump discontinuities in the flux function. Together, the results in this work indicate that the method is consistent, robust, and naturally extensible to more general configurations.