Titlebook: Numerical Methods for Conservation Laws; Randall J. LeVeque Textbook 1992Latest edition Springer Basel AG 1992 CFL condition.average.compa

hypotension

The Derivation of Conservation Lawsdynamics problem, for example flow in a tube where properties of the gas such as density and velocity are assumed to be constant across each cross section of the tube. Let x represent the distance along the tube and let . be the density of the gas at point x and time ..

Inclement

洞察力

Shocks and the Hugoniot Locus eigenvalues λ.(.) <…< λ.(.) and hence linearly independent eigenvectors. We choose a particular basis for these eigenvectors, {.{.)}., usually chosen to be normalized in some manner, e.g. ∥.(itu})∥ ≡ 1.

邪惡的你

The Riemann problem for the Euler equationst the details are messier. Instead, I will concentrate on discussing one new feature seen here, contact discontinuities, and see how we can take advantage of the linear degeneracy of one field to simplify the solution process for a general Riemann problem. Full details are available in many sources, for example [11], [77], [97].

附錄

978-3-7643-2723-1Springer Basel AG 1992

兇殘

陶醉

High Resolution Methods numerical dissipation in monotone methods. Some dissipation is obviously needed to give nonoscillatory shocks and to ensure that we converge to the vanishing viscosity solution, but monotone methods go overboard in this direction.

Asymptomatic

Some Scalar Examplescase of gas dynamics, with gas molecules taking the place of cars. This application is discussed in much more detail in Chapter 3 of Whitham[97]. The second example (two phase flow) shows what can happen when . is not convex.

Generosity

Semi-discrete Methods equations in time, called the “semi-discrete equations”. We then discretize in time using any standard numerical method for systems of ordinary differential equations. This approach of reducing a PDE to a system of ODEs, to which we then apply an ODE solver, is often called the method of lines.

Irascible

Nonlinear Stabilityas been completely successful only for scalar problems. For general systems of equations with arbitrary initial data no numerical method has been proved to be stable or convergent, although convergence results have been obtained in some special cases (. [20], [50], [53]).