TABLE OF CONTENTS Numerical schemes which do not involve the sign of the characteristic speeds in the discretization of the spatial derivatives can be classified as centered schemes. In contrast to the first-order upwind discretization, centered schemes make use of points from the left and the right of the center of the stencil (Fig. 15.1). Examples of centered schemes include the Lax-Friedrichs [322], Lax-Wendroff [321], Toro's first order centered scheme (FORCE) [544], and variants of second and third-order nonoscillatory schemes for hyperbolic conservation laws by Tadmor and collaborators [281, 279, 343, 357, 396].
We consider the one-dimensional, linear advection equation U t + ? U x = 0. Using centered discretization in space and first-order explicit discretization in time, we obtain
Performing von Neumann stability analysis using the solution U i n = E ne ?ki?x, where E n is the amplitude, k is the wave number [1], and ? = 
is the unit complex number, we obtain
The stability requirement is E n ? 1. However,
thus the discretization (15.1) is unconditionally unstable.
The Lax-Friedrichs scheme [322] aims to rectify the above problem by replacing U i n by ( U i ? 1 n + U i +1 n)/2. The linear advection equation is written as
Equation (15.3) can also be written
The last...
