Number of grid points per wavelength

u_t + u_x = 0 u(x, 0) = sin (Lπx) 0<x<1

Method1: Upwind(1st order) Method2: Lax-Wendroff(2nd order) Method3: Traditional 4th order Method4: Compact schemes (4th order)

Upwind L=2 h error grid points per wavelength 1/ / / / / / /

Upwind L=4 h error grid points per wavelength 1/ / / / / / /

Lax-Wendroff L h error grid points per wavelength 2 1/ / / /

Traditional 4th order L h error grid points per wavelength 2 1/ / / /

Compact schemes L h error grid points per wavelength 2 1/ / / /

High Order Schemes for Resolving Waves: Number of Points per Wavelength

First derivative f(x)=sin(kx) TEk=c*(Δx)^(p-1)*k^p p is the order of numerical scheme TEk= c*(1/N)^(p-1)*k^p, Δx=1/N If k changes to m, and we want TE to be unchanged. TEk=TEm (1/N)^(p-1)*k^p= (1/a*N)^(p-1)*m^p, a=(m/k)^(p/(p-1)).

If p>>0, then a~m/k. So, if m=2k, then the number of grid points is also doubled since a~2. But if p is small, say p=2,then when m=2k, the number of grid points should be multiplied by 4 to insure that TE is unchanged.

Higher derivative (1/N)^(p-q)*k^p=(1/a*N)^(p-q)*m^p, a=(m/k)^(p/(p-q)).

1 st order scheme Scheme N error IC UW sin(x) UW sin(2x) L-F sin(x) L-F sin(2x) a=2^2=4

2 nd order scheme Scheme N error IC FD sin(x) FD sin(2x) L-W sin(x) L-W sin(2x) a=2^(3/2)=2.83

4 th order scheme Scheme N error IC FD sin(x) FD sin(2x) FDC sin(x) FDC sin(2x) a=2^(5/4)=2.34

6 th order, a=2^(7/6)= th order, a=2^(9/8)=2.18

Upwind: L changes from 2 to 4, under the same error, 1/h multiplied by 4. grid points per wavelength multiplied by 2. L-W: L=2, 4, 8, 16, under the same error. 1/h multiplied by 2.83 grid points per wavelength multiplied by 1.42

Traditional 4th order L=2, 4, 8, 16, under the same error. 1/h multiplied by 2.37 grid points per wavelength multiplied by 1.19 Compact schemes L=2, 4, 8, 16, under the same error. 1/h multiplied by 2.37 grid points per wavelength multiplied by 1.19