**An Unconditionally Stable Implicit Difference Scheme for the Hydrodynamical Equations**

**Local PDF:** ADA385042.pdf

AD Number: ADA385042

Subject Categories: EXPLOSIONS

Corporate Author: LOS ALAMOS SCIENTIFIC LAB ALBUQUERQUE NM

Title: An Unconditionally Stable Implicit Difference Scheme for the
Hydrodynamical Equations

Personal Authors: Turner, James; Wendroff, Burton

Report Date: 15 APR 1964

Pages: 43 PAGES

Report Number: LA-3007

Contract Number: W-7405-ENG-36

Monitor Acronym: XJ

Monitor Series: AEC

Descriptors: *SHOCK WAVES, *EXPLOSIONS, *HYDRODYNAMIC CODES, NUMERICAL
ANALYSIS, FINITE DIFFERENCE THEORY, DISCONTINUITIES, NONLINEAR ALGEBRAIC
EQUATIONS, RAREFACTION.

Identifiers: NEWTON METHOD, COURANT CONDITION

Abstract: We solve two hydrodynamical problems. The first involves **a** shock
wave, **a** contact discontinuity, and **a** rarefaction wave using an unconditionally
stable finite difference scheme. The Courant condition is satisfied everywhere
except in one zone behind the shock, where it is violated by factors of 10 and
100. The nonlinear difference equations are solved by Newton's method. The total
number of Newton iterations to get to **a** certain time is apparently independent
of the degree to which the normal stability condition is violated in the one
zone. The second problem involves two rarefaction waves moving in opposite
directions. One wave moves in **a** region where the Courant condition is violated
by **a** factor of approximately two. The other wave moves in **a** region where the
Courant condition is satisfied. Numerical results are compared with the
analytical solution. An examination of several runs indicates one explicit time
step is about five times as fast as one implicit time step. Therefore, use of
the implicit method is indicated when the Courant condition is violated by **a**
factor of 5 or more.

Limitation Code: APPROVED FOR PUBLIC RELEASE

Source Code: 394961

Citation Creation Date: 03 JAN 2001