Theoretical and numerical structure for reacting shock waves
SIAM Journal on Scientific and Statistical Computing
The generalized Riemann problem for reactive flows
Journal of Computational Physics
A study of numerical methods for hyperbolic conservation laws with stiff source terms
Journal of Computational Physics
Non-oscillatory central differencing for hyperbolic conservation laws
Journal of Computational Physics
Local error estimates for discontinuous solutions of nonlinear hyperbolic equations
SIAM Journal on Numerical Analysis
High-resolution conservative algorithms for advection in incompressible flow
SIAM Journal on Numerical Analysis
One-dimensional transport equations with discontinuous coefficients
Nonlinear Analysis: Theory, Methods & Applications
Journal of Computational Physics
Mathematics of Computation
The random projection method for hyperbolic conservation laws with stiff reaction terms
Journal of Computational Physics
A Modified Fractional Step Method for the Accurate Approximation of Detonation Waves
SIAM Journal on Scientific Computing
Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
The Random Projection Method for Stiff Detonation Capturing
SIAM Journal on Scientific Computing
Finite-Volume-Particle Methods for Models of Transport of Pollutant in Shallow Water
Journal of Scientific Computing
Modified Optimal Prediction and its Application to a Particle-Method Problem
Journal of Scientific Computing
A velocity--diffusion method for a Lotka--Volterra system with nonlinear cross and self-diffusion
Applied Numerical Mathematics
Convergence of a Particle Method and Global Weak Solutions of a Family of Evolutionary PDEs
SIAM Journal on Numerical Analysis
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This paper is devoted to a practical implementation of deterministic particle methods for solving transport equations with discontinuous coefficients and/or initial data, and related problems. In such methods, the solution is sought in the form of a linear combination of the delta-functions, whose positions and coefficients represent locations and weights of the particles, respectively. The locations and weights of the particles are then evolved in time according to a system of ODEs, obtained from the weak formulation of the transport PDEs.The major theoretical difficulty in solving the resulting system of ODEs is the lack of smoothness of its right-hand side. While the existence of a generalized solution is guaranteed by the theory of Filippov, the uniqueness can only be obtained via a proper regularization. Another difficulty one may encounter is related to an interpretation of the computed solution, whose point values are to be recovered from its particle distribution. We demonstrate that some of known recovering procedures, suitable for smooth functions, may fail to produce reasonable results in the nonsmooth case, and discuss several successful strategies which may be useful in practice. Different approaches are illustrated in a number of numerical examples, including one-and two-dimensional transport equations and the reactive Euler equations of gas dynamics.