Viscosity Solutions and Convergence of Monotone Schemes for Synthetic Aperture Radar Shape-from-Shading Equations with Discontinuous Intensities

An Engquist–Osher-Type Scheme for Conservation Laws with Discontinuous Flux Adapted to Flux Connections

“…We mention here briefly flow in porous media [15], sedimentation processes [5,13,14], and traffic flow on a highway [57,17]. They also arise in radar shape-from-shading problems [44] and as building blocks in numerical methods for Hamilton-Jacobi equations [21] based on dimensional splitting. In view of their applications, there is great demand for accurate, efficient, and, at the same time, easy-to-implement numerical methods for conservation laws with discontinuous coefficients.…”

The ${l}^1$-Error Estimates for a Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials

“…For the discontinuous coefficient case, the convergence of numerical schemes were widely studied [6,9,14,15,4,17,25,19,21,7,26,27,16,13,1,20]. However, in most cases the convergence rate estimates for numerical schemes were not studied.…”

Convergence of an Immersed Interface Upwind Scheme for Linear Advection Equations with Piecewise Constant Coefficients II: Some Related Binomial Coefficients Inequalities

“…In contrast, for hyperbolic equations with singular coefficients, or conservation laws with discontinuous flux functions, the convergence rate results for numerical methods are much less studied, although many authors have studied the convergence of the numerical methods. The convergence studies include the convergence of a front tracking method for conservation laws with discontinuous flux functions [4], the convergence of front tracking schemes [3,5,18,19], the Lax-Friedrichs scheme [17] and convergence rate estimates for Godunov's and Glimm's methods [21,28] for the resonant systems of conservation laws, the convergence of monotone schemes for synthetic aperture radar shape-from-shading equations with discontinuous intensities [23], the convergence of a class of finite difference schemes for the linear conservation equation and the transport equation with discontinuous coefficients [6], the convergence of a difference scheme, based on Godunov or Engquist-Osher flux, for scaler conservation laws with a discontinuous convex flux [30] and the extension to the nonconvex flux [31], the convergence of an upwind difference scheme of Engquist-Osher type for degenerate parabolic convection-diffusion equations with a discontinuous coefficient [16], the convergence of a relaxation scheme for conservation laws with a discontinuous coefficient [15], the convergence of Godunov-type methods for conservation laws with a flux function discontinuous in space [1], the convergence of upwind difference schemes of Godunov and Engquist-Osher type for a scalar conservation law with indefinite discontinuities in the flux function [22]. In the above cases, except for the resonant systems of conservation laws, convergence rates for numerical methods were not studied.…”