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

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

• Consider the following system of two first order wave equations:

$$f_t+u{g}_x=0\stackrel{\partial /\partial t}{\to }{f}_{tt}+u{g}_{xt}=0$$

$$g_t+u{f}_x=0\underset{\ast u,\text{ swap xt order}}{\overset{\partial /\partial x}{\to }}u{g}_{tx}+u^2{f}_{xx}=0$$

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

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

• 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

$$f_t+u{f}_x=0,$$

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

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

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

    *     O     *
    
    O     O     O

Lax

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

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

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

$$f_t+u{f}_x=0.$$

• FTCS:

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

• Taylor series for $f$’s:

$$f_i^{n+1}=f_i^n+\mathrm{\Delta }t{f}_t+\frac{1}{2}\mathrm{\Delta }{t}^2{f}_{tt}+\dots ,$$ $$f_{i-1}^n=f_i^n-\mathrm{\Delta }x{f}_x+\frac{1}{2}\mathrm{\Delta }{x}^2{f}_{xx}+\dots ,$$ $$f_{i+1}^n=f_i^n+\mathrm{\Delta }x{f}_x+\frac{1}{2}\mathrm{\Delta }{x}^2{f}_{xx}+\dots ,$$

• Substitute this into FTCS to get the MDE:

$$f_t+u{f}_x=-\frac{1}{2}\mathrm{\Delta }t{f}_{tt}+(\text{higher order terms}).$$

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

$$\frac{\partial }{\partial t}(f_t=-u{f}_x)\to f_{tt},$$

$$=-u{f}_{tx}=-u(f_t{)}_x,$$

$$=u^2{f}_{xx}$$

• In the last step, we used $f_t=-u{f}_x$.
• Hence, the MDE is:

$$f_t+u{f}_x=\underset{\text{error term}}{\underbrace{-\frac{1}{2}\mathrm{\Delta }t{u}^2{f}_{xx}}}.$$

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

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

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

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

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

    *     O     *
    
    O     O     O

Note

• The error terms are now $f_{ttt}$ and $f_{xxx}$ $\to$ 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

$$\frac{\partial f}{\partial t}=-u\frac{\partial f}{\partial x}=\text{RHS}=F.$$

$$\partial f=F\partial t\stackrel{\int }{\to }{f}_i^{n+1}-f_i^n=\int Fdt=\bar{F}dt,$$

where

$$\bar{F}=\frac{1}{\mathrm{\Delta }t}{\int }_t^{t+\mathrm{\Delta }t}Fdt.$$

• $F$ is the slope in time of $f$, so $\bar{F}$ is the average slope.

• Approximate this average slope as $$\bar{F}=\frac{1}{2}(F^n+F^{n+1}).$$

• Hence, $$f_i^{n+1}=f_i^n+\frac{\mathrm{\Delta }t}{2}(F_i^n+F_i^{n+1}).$$

• 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: $f_x=(f_{i+1}-f_i)/\mathrm{\Delta }x$. Denote this as ${}^{+}{F}_i^{n+1}.$
  • When discretizing $F_i^{n+1}$ we use a backward difference: $f_x=(f_i-f_{i-1})/\mathrm{\Delta }x$. 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.

• $\mathcal{O}(\mathrm{\Delta }{t}^2)$

• $\mathcal{O}(\mathrm{\Delta }{x}^2)$

• Stable for $c=\frac{u\mathrm{\Delta }t}{\mathrm{\Delta }x}\le 1$

• Consistent

• Stencil:

    O     O     *
    
    *     O     O

Upwind method

• We have seen this already.

• For positive $u$ we have $$f_i^{n+1}=f_i^n-c(f_i^n-f_{i-1}^n).$$

• $\mathcal{O}(\mathrm{\Delta }t)$

• $\mathcal{O}(\mathrm{\Delta }x)$

• Stable for $c=\frac{u\mathrm{\Delta }t}{\mathrm{\Delta }x}\le 1$

• 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 $c=1$.

• $\mathcal{O}(\mathrm{\Delta }{t}^2)$

• $\mathcal{O}(\mathrm{\Delta }{x}^2)$

• Stable for $c=\frac{u\mathrm{\Delta }t}{\mathrm{\Delta }x}\le 2$

• Consistent

• Stencil:

    *     *     O     *
    
    O     O     O     *

BTCS

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

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

    O     O     O
   
    *     O     *

Methods Summary

Method	Recommend	$\mathcal{O}(\mathrm{\Delta }t)$	$\mathcal{O}(\mathrm{\Delta }x)$	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