Hyperbolic equations 1

Consider equations with only convection

$$\frac{\partial f}{\partial t} + u\frac{\partial f}{\partial x} = 0.$$

Assume constant $u$.
This is a linear PDE.
Exact solution

$$f(x,t) = f_0(x-ut),$$

where $f_0(x)$ is the initial $f$ profile. That is, we just shift the domain. This is consistent with this PDE being a wave equation, where the solution advects across the domain without changing shape.

Note

The second order wave equation is given by

$$\frac{\partial^2f}{\partial^2 t} = u^2\frac{\partial^2f}{\partial^2 x}.$$

Consider the following system of two first order wave equations:

$$f_t + ug_x = 0 \xrightarrow{\partial/\partial t} f_{tt} + ug_{xt} = 0$$$$g_t + uf_x = 0 \xrightarrow[*u,\text{ swap xt order}]{\partial/\partial x} ug_{tx} + u^2f_{xx} = 0$$

Now subtract the second equation from the first to recover the second order wave equation:

$$f_{tt} - u^2f_{xx} = 0.$$

This shows that the second order wave equation is equivalent to a system of two first order wave equations.
Here, we’ll focus on the first order wave equation.

Methods

FTCS
Lax
Lax Wendroff
MacCormack
Upwind
BTCS

FTCS

$$f_t + uf_x = 0,$$$$\frac{f_i^{n+1}-f_i^n}{\Delta t} + u\frac{f_{i-1}^n-f_{i+1}^n}{2\Delta x} = 0,$$$$f_i^{n+1} = f_i^n - \frac{c}{2}(f_{i+1}^n-f_{i-1}^n).$$

$c=u\Delta t/\Delta x$
$\mathcal{O}(\Delta t)$
$\mathcal{O}(\Delta x^2)$
unconditionally unstable
Consistent
Stencil:

    *     O     *
    
    O     O     O

Lax

Fix the instability.
Instead of $f_i^n$ use $\frac{1}{2}(f_{i-1}^n + f_{i+1}^n).$

$$f_i^{n+1} = \frac{1}{2}(f_{i-1}^n+f_{i+1}^n) - \frac{u}{2}(f_{i+1}^n-f_{i-1}^n).$$

$\mathcal{O}(\Delta t)$
$\mathcal{O}(\Delta x^2)$
Stable for $c=\frac{u\Delta t}{\Delta x}\le 1$
- $c\le 1$ is the Courant Friedrichs Lewy (CFL) stability criterion
Inconsistent

    *     O     *
    
    O     *     O

Lax Wendroff

Fix the FTCS method
Consider the modified differential equation: MDE
- When we solve PDEs as finite difference equations (FDEs), the FDE has truncation error (from terms we ignore in the Taylor series).
- A Taylor series can be used to show the PDE that is equivalent to the FDE we are solving. This equivalent PDE is called the modified differential equation (MDE).
PDE:

$$f_t + uf_x = 0.$$

FTCS:

$$\frac{f_i^{n+1}-f_i^n}{\Delta t} + \frac{c}{2\Delta t}(f_{i-1}^n-f_{i+1}^n) = 0.$$

Taylor series for $f$’s:

$$ f_i^{n+1} = f_i^n + \Delta t f_t + \frac{1}{2}\Delta t^2f_{tt} + \ldots,$$

$$ f_{i-1}^{n} = f_i^n - \Delta x f_x + \frac{1}{2}\Delta x^2f_{xx} + \ldots,$$

$$ f_{i+1}^{n} = f_i^n + \Delta x f_x + \frac{1}{2}\Delta x^2f_{xx} + \ldots,$$

Substitute this into FTCS to get the MDE:

$$f_t + uf_x = -\frac{1}{2}\Delta tf_{tt} + (\text{higher order terms}).$$

Now, the $f_{tt}$ term in this equation can be replaced with $u^2f_{xx}$ as follows:

$$\frac{\partial}{\partial t}(f_t = -uf_x)\rightarrow f_{tt},$$$$=-uf_{tx}=-u(f_t)_x,$$$$=u^2f_{xx}$$

In the last step, we used $f_t = -uf_x$.
Hence, the MDE is:

$$f_t + uf_x = \underbrace{-\frac{1}{2}\Delta t u^2f_{xx}}_{\text{error term}}.$$

Now, subtract off the error term from the RHS of FTCS:

$$\frac{f_i^{n+1}-f_i^n}{\Delta t} + \frac{c}{2\Delta t}(f_{i+1}^n-f_{i-1}^n) = \frac{1}{2}\Delta tu^2f_{xx}.$$

Finally, discretize the $f_{xx}$ term and rearrange:

$$f_i^{n+1} = f_i^n - \frac{c}{2}(f_{i+1}^n-f_{i-1}^n) + \frac{1}{2}c^2(f_{i-1}^n - 2f_i^n + f_{i+1}^n).$$

$\mathcal{O}(\Delta t^2)$
$\mathcal{O}(\Delta x^2)$
Stable for $c=\frac{u\Delta t}{\Delta x}\le 1$
Consistent
Stencil:

    *     O     *
    
    O     O     O

Note

The error terms are now $f_{ttt}$ and $f_{xxx}$ $\rightarrow$ third order derivatives in the error terms.
- Error terms that have even derivatives (like $f_{xx}$) are diffusive and cause unphysical smoothing in the solution.
- Error terms that have odd derivatives (like $f_{xxx}$) are dispersive and cause unphysical wiggles in the solution.

Question

Normally, diffusive terms help stability, so why is it that the error term in the FTCS method above (which is a second order, diffusive term) is the cause of the instability?
- What does this term do physically?
- Look at the term and consider its form.

MacCormack

Was and is widely used
Used for systems of equations, nonlinear equations
Similar to Crank-Nicolson

$$\frac{\partial f}{\partial t} = -u\frac{\partial f}{\partial x} = \text{RHS} = F.$$$$\partial f = F\partial t \xrightarrow{\int} f_i^{n+1}-f_i^n = \int Fdt = \overline{F}dt,$$

where

$$\overline{F} = \frac{1}{\Delta t}\int_t^{t+\Delta t}Fdt.$$

$F$ is the slope in time of $f$, so $\overline{F}$ is the average slope.
Approximate this average slope as
$$\overline{F} = \frac{1}{2}(F^n+F^{n+1}).$$
Hence,
$$f_i^{n+1} = f_i^n +\frac{\Delta t}{2}(F_i^n + F_i^{n+1}).$$
If we take an explicit Euler step to $\tilde{f}_i^{n+1}$, then we can evaluate $F_i^{n+1}$ there. Then we can evaluate the above equation explicitly.
- This would be the equivalent of the Modified Euler method for PDEs.
We do this, but
- When discretizing $F_i^n$ we use a forward difference: $f_x = (f_{i+1}-f_i)/\Delta x$. Denote this as $^+F_i^{n+1}.$
- When discretizing $F_i^{n+1}$ we use a backward difference: $f_x = (f_{i}-f_{i-1})/\Delta x$. Denote this as $^-F_i^{n+1}.$
Summary:
- Predictor:
$$\tilde{f}_i^{n+1}=f_i^n - c(f\_{i+1}^n-f_i^n),$$
- Corrector:
$$f_i^{n+1} = f_i^n - \frac{1}{2}[c(f\_{i+1}^n-f_i^n) + c(\tilde{f}_i^{n+1}-\tilde{f}\_{i-1}^{n+1})].$$
- This second equation can be written as
  $$f_i^{n+1} = \frac{1}{2}[f_i^n + \tilde{f}_i^{n+1} - c(\tilde{f}_i^{n+1} - \tilde{f}\_{i-1}^{n+1})].$$
For the 1-D linear wave equation, this method is the same as the Lax Wendroff method.
$\mathcal{O}(\Delta t^2)$
$\mathcal{O}(\Delta x^2)$
Stable for $c=\frac{u\Delta t}{\Delta x}\le 1$
Consistent
Stencil:

    O     O     *
    
    *     O     O

Upwind method

We have seen this already.
For positive $u$ we have
$$ f_i^{n+1} = f_i^n - c(f_i^n - f_{i-1}^n).$$
$\mathcal{O}(\Delta t)$
$\mathcal{O}(\Delta x)$
Stable for $c=\frac{u\Delta t}{\Delta x}\le 1$
Consistent
Stencil:

    *     O     *
    
    O     O     *

Second order upwind

If we use our old second order one-sided finite difference approximation for $f_x$, the result is unconditionally unstable.
Hoffman gives an alternative:

$$f_i^{n+1} = f_i^n - c(f_i^n - f_{i-1}^n) - \frac{c(1-c)}{2}(f_i^n - 2f_{i-1}^n + f_{i-2}^n).$$

This equation is exact for the linear 1D wave equation for $c=1$.
$\mathcal{O}(\Delta t^2)$
$\mathcal{O}(\Delta x^2)$
Stable for $c=\frac{u\Delta t}{\Delta x}\le 2$
Consistent
Stencil:

    *     *     O     *
    
    O     O     O     *

BTCS

$$\frac{f_i^{n+1}-f_i^n}{\Delta t} + \frac{u}{2\Delta x}(f_{i-1}^{n+1}-f_{i-1}^{n+1}) = 0.$$

$\mathcal{O}(\Delta t)$
$\mathcal{O}(\Delta x^2)$
Unconditionally stable
Consistent
Unphysical: $\infty$ propagation speed of information, unlike the PDE.
- The wave lags behind the real answer.
Stencil:

    O     O     O
   
    *     O     *

Methods Summary

Method	Recommend	$\mathcal{O}(\Delta t)$	$\mathcal{O}(\Delta x)$	Approach	Stability	Comment
FTCS	X	1	2	direct, forward in time, central in space	NO	doesn’t work
Lax	X	1	2	average $f_i^n$	$c\le 1$	inconsistent
Lax Wendroff	okay	2	2	fix low order error $f_{tt}$ term	$c\le 1$	hard for nonlinear, systems, multi-D
MacCormack	good	2	2	time adv. with avg. RHS slope:	$c\le 1$	good for nonlinear, systems, multi-D
Upwind	poor	1	1	very diffusive	$c\le 1$	diffusive, low order
2nd Order Upwind	okay	2	2	3 point stencil	$c\le 2$	oscillatory
BTCS	poor	1	2	implicit	$c\le 2$	unphysical info. speed