MATH761Oct 15, 2018

1. Read sections 1-3 of the paper “Group velocity in finite difference schemes”, published in SIAM Review, Vol. 24, No. 2, April 1982, authored by Lloyd N. Trefethen, available at https://people.maths.ox.ac.uk/trefethen/publication/PDF/1982_6.pdf. Present a three-paragraph summary of the main points of these paper sections.

   Solution.

   1. Dispersion relation and group velocity. A monochromatic wave $u(t,x)=\exp \left[i(\omega t-\xi x)\right]$ propagates at phase speed $c\left(\xi \right)=\omega \left(\xi \right)/\xi$, but a superposition of multiple monochromatic waves, i.e., a wave packet with components $F\left(\xi \right)$,

      ${\displaystyle f(t,x)=\underset{-\infty }{\overset{\infty }{\int }}F\left(\xi \right)\exp [it(\omega \left(\xi \right)-\frac{\xi x}{t})]d\xi ,}$

      has asymptotic behavior dominated by the stationary phase condition

      ${\displaystyle \frac{d}{d\xi }(\omega \left(\xi \right)-\frac{\xi x}{t})=0\Rightarrow \frac{d\omega }{d\xi }=\frac{x}{t}=C\left(\xi \right),}$

      and travels at the group speed $C\left(\xi \right).$ The relationship $\omega \left(\xi \right)$ obtained by substituting a monochromatic into an equation is the dispersion relationship, and the exact dispersion relationship $\omega \left(\xi \right)=\xi$ for the wave equation $u_t+u_x=0$ is approximated by the finite difference dispersion relationships in Table 1.

      Scheme Dispersion relation Leapfrog $\sin \omega k=\nu \sin \xi h$ Crank-Nicolson $2\tan (\frac{1}{2}\omega k)=\nu \sin \xi h$ Leapfrog fourth order $\sin \omega k=\frac{4}{3}\nu \sin \xi h-\frac{1}{3}\nu \sin 2\xi h$

      Table 1. Finite difference dispersion relationships, $\nu =1\cdot k/h$ is the CFL number

   2. Pulses, wave packets and wave fronts. The finite difference approximation of wave packet propagation exhibits errors in both phase speed and group speed. Dispersion occurs if there is a significant spread in the wavenuber of the packet.

   3. Parasites, interfaces and mesh refinement. The errors in finite difference approximations of group velocity leads to non-physical (parasitic) waves, especially at boundaries where the physical wave speed changes, e.g., between two boundaries.

2. Consider the longitudinal plane wave ${𝒖}_i(t,x)={𝒆}_x\exp \left[i(\omega t-\xi x)\right]$, and a Hookean elastic medium in which conservation of momentum is stated as

   ${\displaystyle \rho {\partial }^2𝒖/\partial t^2=(\lambda +2\mu )\nabla (\nabla \cdot 𝒖)-\mu \nabla \times (\nabla \times 𝒖)}$

   (1)

   where:

   • $𝒖(t,𝒙)$ is the displacement [m]

   • $\rho$ is the density [kg/m${}^3$]

   • $\lambda ,\mu$ are the bulk, shear elastic moduli [N/m${}^2$].

   1. Is the longitudinal plane wave a solution of the equation of motion (1)?

      Solution. Compute

      ${\displaystyle {\partial }^2𝒖/\partial t^2=-{\omega }^2{𝒖}_i(t,x),\nabla \cdot {𝒖}_i=-i\xi \exp \left[i(\omega t-\xi x)\right],\nabla (\nabla \cdot 𝒖)=-{\xi }^2{𝒖}_i(t,x),\nabla \times 𝒖=0,}$

      and obtain a condition for the longitudinal wave to be a solution

      ${\displaystyle -\rho {\omega }^2=-(\lambda +2\mu ){\xi }^2.}$

      (2)

   2. At what speed does the wave propagate?

      Solution. From (2) obtain the phase speed

      ${\displaystyle c=\frac{\omega }{\xi }=\pm \sqrt{\frac{\lambda +2\mu }{\rho }}.}$

      (3)

   3. At an interface with another elastic medium $({\lambda }^{\text{'}},{\mu }^{\text{'}})$ the incident wave ${𝒖}_i$ produces a reflected ${𝒖}_r$ and transmitted wave ${𝒖}_t$. Derive formulas for the reflection and transmission coefficients

      ${\displaystyle R=\frac{||{𝒖}_r||}{||{𝒖}_i||},T=\frac{||{𝒖}_t||}{||{𝒖}_i||}}$

      Solution. The reflected, transmitted waves are $u_r=R\exp \left[i(\omega t+\xi x)\right]$, $u_t=T\exp \left[i(\omega t-{\xi }^{\text{'}}x)\right]$. At the interface $x=0$ impose the conditions:

      Kinematic condition

      Same displacement on the two sides of the boundary:

      ${\displaystyle u_i(t,x)+u_r(t,x)=u_t(t,x)\Rightarrow 1+R=T}$

      Dynamic condition

      In this one-dimensional case $\rho u_{tt}={\partial }_x\left[(\lambda +2\mu )u_x\right]={\partial }_x\left[\sigma \right]$, with $\sigma$ the normal stress, that has to be the same on the two sides of the boundary

      ${\displaystyle (\lambda +2\mu ){\partial }_x[u_i(t,x)+u_r(t,x)]=({\lambda }^{\text{'}}+2{\mu }^{\text{'}}){\partial }_x{u}_t(t,x)\Rightarrow }$ ${\displaystyle (\lambda +2\mu )\xi (-1+R)=-({\lambda }^{\text{'}}+2{\mu }^{\text{'}}){\xi }^{\text{'}}T.}$

      Since $c=\sqrt{(\lambda +2\mu )/\rho }=\omega /\xi$, $c^{\text{'}}=\sqrt{({\lambda }^{\text{'}}+2{\mu }^{\text{'}})/{\rho }^{\text{'}}}=\omega /{\xi }^{\text{'}}$, the system is stated in terms of the impedance $Z=\rho c$, as

      ${\displaystyle \begin{array}{rcl}1+R& =& T\\ Z(-1+R)& =& -Z^{\text{'}}T\end{array}}$

      leading to

      ${\displaystyle R=\frac{Z-Z^{\text{'}}}{Z+Z^{\text{'}}},T=\frac{2Z}{Z+Z^{\text{'}}}.}$

3. Consider a computational domain for (1) discretized by the second-order leap-frog scheme (LF), extending to distance $L$ of each side of $x=0$. The subdomain $x<0$ is discretized with step size $h$, the subdomain $x>0$ is discretized with step size $2h$. At the left edge $x=-L$ an incoming plane wave $u_i(t,x)$ is entering the domain. The physics of the problem predicts unattenuated, non-dispersive passage of the wave, but the analysis within the Trefethen paper shows that the discrete scheme will induce both attenuation and dispersion.

   1. Compute the waveform at the interface $u_i(t,0_{-})$ obtained after $u_i(t,-L)$ has propagated through the grid with step size $h$. Pay attention to the $\xi /L$ ratio.

      Solution. The dispersion relation for LF is $\sin \omega k=\nu \sin \xi h$. A monochromatic wave will propagate with phase velocity

      ${\displaystyle c_{\mathrm{LF}}=\frac{\omega }{\xi }\cong 1-\delta c_{\mathrm{LF}},\delta c_{\mathrm{LF}}=\frac{1-{\nu }^2}{6}{(\xi h)}^2,}$

      arriving $\delta c_{\mathrm{LF}}/L$ time units later than the physical wave, and no change in shape, i.e., dissipation is $𝒪\left(h^3\right)$, and the only effect of dispersion is a time delay.

   2. Compute the reflection and transmission coefficients at the interface between the two grids

      ${\displaystyle R=\frac{\left|u_r(t,0_{-})\right|}{\left|u_i(t,0_{-})\right|},T=\frac{\left|u_t(t,0_{+})\right|}{\left|u_i(t,0_{-})\right|}.}$

      Solution. The same physical principles apply, so the only change is that the impedance is now $Z=\rho (c+\delta c_{\mathrm{LF}})$.

   3. Determine the ratio of change in material properties that would produce the same reflection and transmission coefficients for the exact wave equation as that produced by the grid spacing jump from $h$ to $2h$ for the discretized wave equation.

      Solution.