A stochastic particle method for the solution of a 1D viscous scalar conservation law in a bounded interval (Q2724978)

A viscous scalar conservation law with Dirichlet boundary condition is interpreted as a nonlinear martingale problem for a stochastic process with reflection at the boundary. The authors prove the uniqueness and the existence of the solution of the martingale problem. The latter is done by exhibiting a system of \(N\) interacting particles whose empirical measure converges to the unique solution of the martingale problem as \(N\to\infty\). This provides a possibility of approximating the solution of the viscous conservation law using simulation of a particle system. The system is discretized in time using the Euler-Lépinge scheme that yields the convergence rate of \({\mathcal O}(\Delta t+1/\sqrt N)\), where \(\Delta t\) denotes the time step. Numerical results given in the paper confirm this theoretical rate.