A Class of Positive Semi-discrete Lagrangian–Eulerian Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

Convergence, bounded variation properties and Kruzhkov solution of a fully discrete Lagrangian–Eulerian scheme via weak asymptotic analysis for 1D hyperbolic problems

“…Roughly speaking, our method can be viewed as the desirable middle term between Godunov‐upwinding schemes and central differencing schemes , but with significant differences. As in the Lagrangian–Eulerian schemes presented in previous works for local hyperbolic problems (see [6–9]), the local conservation is obtained by integrating the conservation law over the region in the space–time domain where the conservation of mass flux takes place, preserving the key ingredients: (1) the evolution Lagrangian step is based on the improved concept of the no‐flow curves [8–11] and using the same mass conservation property as in [27], in which the numerical approximation in the space–time control volume at the time $$$ {t}^n $$$ is evolved to time $$$ {t}^{n+1} $$$ and (2) a Eulerian remap (second step), where the current approximation is projected over the original grid. The fully discrete Lagrangian–Eulerian scheme formulation discussed here is based on the new and substantial improvement interpretation of the integral tube , here under name no‐flow curves [10, 11].…”

“…To handle this situation, we can easily apply a flux separation strategy (see, [11] for more details), which transforms the original hyperbolic conservation law model equation into a new hyperbolic problem with a modified source term. This strategy has been proven effective on a large class of hyperbolic problems for solving one‐dimensional and multidimensional initial value problems for scalar models and systems of conservation laws [6, 7, 10]. Another feature of the proposed fully discrete Lagrangian–Eulerian scheme is that there is no need to take derivative on the of the term $$$ \frac{H(u)}{u} $$$ in the system of no‐flow Equations (6) either to construct the numerical flux or to the design of the CFL stability condition.…”

“…Roughly speaking, our method can be viewed as the desirable middle term between Godunov-upwinding schemes and central differencing schemes, but with significant differences. As in the Lagrangian-Eulerian schemes presented in previous works for local hyperbolic problems (see [6][7][8][9]), the local conservation is obtained by integrating the conservation law over the region in the space-time domain where the conservation of mass flux takes place, preserving the key ingredients:…”

“…Recently, the Lagrangian-Eulerian approach and the concept of the no-flow curves were generalized for nonlocal problems, as such the fractional conservation laws for nonlinear transport equation with nonlocal velocity [5] and for nonlocal scalar conservation laws on bounded domains and applications in traffic flow [4]. An effective advance of our proposed Lagrangian-Eulerian scheme over Godunov-upwinding type schemes (in case of systems for 1D and/or multidimensional problems [6]) is to be Riemann-solver-free and to avoid (time-consuming) field-by-field type decomposition for tracing the direction of the wind. Besides, the forward (Riemann-solver-free) tracking reduces numerical dissipation.…”

A PINN computational study for a scalar 2D inviscid Burgers model with Riemann data

“…The optimizer is Adam, instead of the LBFGS algorithm in [1]. Figure 2: Reference solution (1) via [3].…”