A generalized numerical scheme for the linear hyperbolic equation $u_t + au_x = 0$ has the following form
$$
\frac{u_j^{n+1} – u_j^n}{\Delta t} + a\frac{u_{j+1}^{n} – u_{j-1}^n}{2\Delta x} – \chi \frac{u_{j+1}^{n} – 2u_{j}^n + u_{j-1}^n}{\Delta x^2} = 0 .
$$
a) Find the values of $\chi$ so that the above scheme is the central-explicit, forward, backward, Lax-Friedrich and Lax-Wendroff scheme.b) What is the order of truncation for each scheme?
c) What is the stability condition for each of these schemes?
d) In what range of $\chi$ the scheme is unconditionally unstable?
e) Classify the schemes in (a) as (i) dissipative, and (ii) dispersive.
ATTEMPT
(a) Notice that if $\chi = 0$, then we obtain the central explicit scheme. If we write the numerical discretization as
$$u_j^{n+1} = u_j^n – \frac{ a \Delta t }{2 \Delta x} (u_{j+1}^n – u_{j-1}^n) + \frac{ \chi \Delta t }{\Delta x^2} (u_{j+1}^n – 2 u_j^n + u_{j-1}^n )$$
And one can see that if $\chi = a \Delta t/2$, then we have the LW scheme. If $\chi = a \Delta x/2$, then we have
$$ u_j^{n+1} = u_j^n + \frac{ a \Delta t }{\Delta x} (u_{j+1}^n – u_j^n) $$
which is forward explicit and if $\chi = – a \Delta x/2$, we obtain backward explicit. Now if $\chi = \frac{ \Delta x^2}{2 \Delta t }$, then we obtain Lax-Friedrichs.
We go to part (c) as (b) is trivial. Perhaps we can combine (c) and (d) in a single problem if we apply the discrete fourier transform:
$$ \hat{u}^{n+1} = \hat{u}^n – r( e^{ij \xi} -e^{-ij \xi } ) \hat{u}^n + \frac{ 2 \chi r }{a \Delta x} (e^{ij \xi} -e^{-ij \xi } -2 ) \hat{u}^n$$
Now, am I on the right track here? Is this the correct approach? As for e), I need some suggestions, how would we approach it?
Best Answer
The proposed values of $\chi$ in a) are correct. One notes that the method is a straightforward finite-difference approximation of the advection-diffusion method $u_t + au_x = \chi u_{xx}$. Introducing the Courant number $r = |a| {\Delta t}/{\Delta x}$, the diffusion constants from a) rewrite as $$ \chi \in |a|\frac{\Delta x}{2} \times \left\lbrace \frac{1}{r},\, 1,\, r,\, 0 \right\rbrace , $$ which correspond to the numerical viscosities of the Lax-Friedrichs, upwind, Lax-Wendroff, and central explicit method, respectively. One notes that if $0<r<1$, then those values of $\chi$ are sorted in decreasing order.
As done in this post, let us assume a perturbation of the form $u_j^n = \xi^n e^{-\text i k j \Delta x}$, where $\xi = \text e^{\text i \omega \Delta t}$. Note that $\omega$ is a complex number, with real part $\omega_R$ and imaginary part $\omega_I$. Injecting this Ansatz in the time-stepping formula, Euler's formulas lead to $$ \xi = 1 + \text i r \sin( k \Delta x) + 2\lambda \left( \cos(k \Delta x) -1 \right) , $$ where $r$ is the Courant number and $$ \lambda = {\chi}\frac{\Delta t}{\Delta x^2} \in \left\lbrace \frac{1}{2},\, \frac{r}{2},\, \frac{r^2}{2},\, 0 \right\rbrace $$ denotes the Fourier number. From this equation, one equates the real and imaginary parts: \begin{aligned} \text{Re}\, \xi &= e^{-\omega_I \Delta t}\cos(\omega_R \Delta t) = 1 + 2\lambda \left( \cos(k \Delta x) -1 \right) ,\\ \text{Im}\, \xi &= e^{-\omega_I \Delta t}\sin(\omega_R \Delta t) = r \sin( k \Delta x) \, . \end{aligned} The stability can be analyzed in terms of the squared amplification factor $|\xi|^2$ which has to be smaller than one. Hence, we must have $$ |\xi|^2 - 1 = 4\lambda^2 - 4 \lambda + r^2 + 4 \lambda (1-2\lambda) Z + (4\lambda^2 - r^2) Z^2 \leq 0 $$ for all $Z = \cos( k \Delta x)$ in $[-1, 1]$. The dispersion is analyzed in terms of the phase velocity ${\omega_R}/{k}$ with ${\omega_R}\Delta t = \arctan\left[{\text{Im}\, \xi}/{\text{Re}\, \xi}\right]$. The attenuation is analyzed in terms of ${\omega_I}\Delta t = -\ln |\xi|$ which is non-negative if $|\xi| \leq 1$ (cf. e.g. (1) p. 182).
(1) E. Godlewski, P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1996. doi:10.1007/978-1-4612-0713-9