Finite Volume Schemes for One-Dimensional Systems

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

“…This condition gives the compact embedding of $$$ BV $$$ functions in $$$ {L}^1 $$$, one can use the Helly's selection theorem to show compactness for the approximations and establish pointwise convergence. After, it is proved that the solution is a weak solution for the scalar equations and, finally, it is proved that the solution satisfies an entropy criterion, the most common used is the Kruzhkov entropy condition, see [29]. In this section, we show the convergence of our scheme using the weak asymptotic theory, which is very suitable to handle with the Eulerian–Lagrangian approach.…”

“…Integrating (42) in x ∈ S 1 , we notice that, due to translations of ±𝜀, there are two-by-two simplifications of the terms of (42), as in Equation (29). Thus, we get…”

“…This condition gives the compact embedding of $$$ BV $$$ functions in $$$ {L}^1 $$$; one can use the Helly's selection theorem to show pointwise convergence of the generated sequence of solutions. After, it is proved that the generated sequence is a weak solution for the scalar equations and, finally, it is proved that the solution satisfies an entropy criterion, the most common used is the Kruzhkov entropy condition, see [29]. The treatment used in the weak asymptotic analysis is similar to the previous steps.…”