Showing papers by "Stanley Osher published in 1981"
TL;DR: In this article, simple upwind finite difference and element methods approximating nonlinear partial differential equations have been developed to approximate general systems of nonlinear hyperbolic conservation laws, nonlinear singular perturbation problems, and particular physical problems involving the equations of compressible fluid dynamics.
Abstract: We have recently, with Bjorn Engquist and various students, developed simple upwind finite difference and element methods approximating nonlinear partial differential equations. These methods are used to approximate (l) general systems of nonlinear hyperbolic conservation laws, (2) nonlinear singular perturbation problems, and (3) particular physical problems involving the equations of compressible fluid dynamics.
79 citations
TL;DR: In this article, a simple difference scheme was proposed to approximate the transport term of the boundary value problem with the same boundary conditions and an arbitrary initial guess and proved convergence to a unique steady state solution as the numerical artificial time approaches infinity.
Abstract: We consider the boundary value problem $\varepsilon y'' - a(y)y' - b(x,y) = F(x), - 1 \leqq x \leqq 1$, for $y( - 1)$ and $y(1)$ given and $b(x,0) \equiv 0$, $b_y (x,y) \geqq 0$. We construct a simple difference scheme approximating $\varepsilon u_{xx} - a(u)u_x - b(x,u) - F(x) = u_t $ for $t \geqq 0, - 1 \leqq x \leqq 1$ with the same boundary conditions and an arbitrary initial guess. As the numerical artificial time approaches infinity, we prove convergence to a unique steady state solution. As the mesh size $\Delta x$ and the parameter $\varepsilon $ approach zero in any order, we prove convergence to the solution of the corresponding O.D.E. We use one-sided differences to approximate the transport term. The method applies equally well to multidimensional analogues. We present numerical results verifying the theory.
58 citations