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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
This discusses figures 8.3.6 to 8.3.8 of Hirsch
Convection Equation is (19.1.6) in Numerical Recipes
Lax-Friedrich's Scheme is called Lax Method -
Equations (19.1.14,15) in Numerical Recipes
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.
The Leapfrog Method is Figure 19.1.5 and
Equation 19.1.30 in Numerical Recipes
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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