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Foil 17 Comparison of Different Methods for Solving Linear Convection Equation

From Introduction to Physics and Numerics of NAS Benchmarks CPSP713 Case studies in Computational Science -- Spring Semester 1996. by Geoffrey C. Fox
1 This discusses figures 8.3.6 to 8.3.8 of Hirsch
2 Convection Equation is (19.1.6) in Numerical Recipes
3 Lax-Friedrich's Scheme is called Lax Method -
4 Equations (19.1.14,15) in Numerical Recipes
5 First Order Upwind Scheme is Equation (19.1.27) and Figure 19.1.4 in Numerical Recipes
  • Note effect of numerical viscosity in Lax and Upwind methods which unphysically damps wave propagation in last two figures.
6 The Leapfrog Method is Figure 19.1.5 and
7 Equation 19.1.30 in Numerical Recipes
8 The Lax-Wendroff Method is Equations (19.1.37,38) and Figure 19.1.7 of Numerical Recipes
  • Both Leapfrog and Lax-Wendroff have formal errors which are second order in time step. Other two methods are only first order accurate.
  • The smaller numerical viscosity is good in figures 8.3.7,8 but causes some unphysical oscillations in figure 8.3.6 for both second order methods
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