Appendix: Useful formulas¶
Finite difference operator notation¶
\[\tag{609}
u'(t_n) \approx
\lbrack D_tu\rbrack^n = \frac{u^{n+\frac{1}{2}} - u^{n-\frac{1}{2}}}{\Delta t}\]
\[\tag{610}
u'(t_n) \approx
\lbrack D_{2t}u\rbrack^n = \frac{u^{n+1} - u^{n-1}}{2\Delta t}\]
\[\tag{611}
u'(t_n) =
\lbrack D_t^-u\rbrack^n = \frac{u^{n} - u^{n-1}}{\Delta t}\]
\[\tag{612}
u'(t_n) \approx
\lbrack D_t^+u\rbrack^n = \frac{u^{n+1} - u^{n}}{\Delta t}\]
\[\tag{613}
u'(t_{n+\theta}) =
\lbrack \bar D_tu\rbrack^{n+\theta} = \frac{u^{n+1} - u^{n}}{\Delta t}\]
\[\tag{614}
u'(t_n) \approx
\lbrack D_t^{2-}u\rbrack^n = \frac{3u^{n} - 4u^{n-1} + u^{n-2}}{2\Delta t}\]
\[\tag{615}
u''(t_n) \approx
\lbrack D_tD_t u\rbrack^n = \frac{u^{n+1} - 2u^{n} + u^{n-1}}{\Delta t^2}\]
\[\tag{616}
u(t_{n+\frac{1}{2}}) \approx \lbrack \overline{u}^{t}\rbrack^{n+\frac{1}{2}}
= \frac{1}{2}(u^{n+1} + u^n)\]
\[\tag{617}
u(t_{n+\frac{1}{2}})^2 \approx \lbrack \overline{u^2}^{t,g}\rbrack^{n+\frac{1}{2}}
= u^{n+1}u^n\]
\[\tag{618}
u(t_{n+\frac{1}{2}}) \approx \lbrack \overline{u}^{t,h}\rbrack^{n+\frac{1}{2}}
= \frac{2}{\frac{1}{u^{n+1}} + \frac{1}{u^n}}\]
\[\tag{619}
u(t_{n+\theta}) \approx \lbrack \overline{u}^{t,\theta}\rbrack^{n+\theta}
= \theta u^{n+1} + (1-\theta)u^n ,\]
\[\tag{620}
\qquad t_{n+\theta}=\theta t_{n+1} + (1-\theta)t_{n-1}\]
Some may wonder why \(\theta\) is absent on the right-hand side of (613). The fraction is an approximation to the derivative at the point \(t_{n+\theta}=\theta t_{n+1} + (1-theta) t_{n}\).
Truncation errors of finite difference approximations¶
\[{u_{\small\mbox{e}}}'(t_n) =
[D_t{u_{\small\mbox{e}}}]^n + R^n = \frac{{u_{\small\mbox{e}}}^{n+\frac{1}{2}} - {u_{\small\mbox{e}}}^{n-\frac{1}{2}}}{\Delta t} +R^n\nonumber,\]
\[\tag{621}
R^n = -\frac{1}{24}{u_{\small\mbox{e}}}'''(t_n)\Delta t^2 + {\cal O}(\Delta t^4)\]
\[{u_{\small\mbox{e}}}'(t_n) =
[D_{2t}{u_{\small\mbox{e}}}]^n +R^n = \frac{{u_{\small\mbox{e}}}^{n+1} - {u_{\small\mbox{e}}}^{n-1}}{2\Delta t} +
R^n\nonumber,\]
\[\tag{622}
R^n = -\frac{1}{6}{u_{\small\mbox{e}}}'''(t_n)\Delta t^2 + {\cal O}(\Delta t^4)\]
\[{u_{\small\mbox{e}}}'(t_n) =
[D_t^-{u_{\small\mbox{e}}}]^n +R^n = \frac{{u_{\small\mbox{e}}}^{n} - {u_{\small\mbox{e}}}^{n-1}}{\Delta t}
+R^n\nonumber,\]
\[\tag{623}
R^n = -\frac{1}{2}{u_{\small\mbox{e}}}''(t_n)\Delta t + {\cal O}(\Delta t^2)\]
\[{u_{\small\mbox{e}}}'(t_n) =
[D_t^+{u_{\small\mbox{e}}}]^n +R^n = \frac{{u_{\small\mbox{e}}}^{n+1} - {u_{\small\mbox{e}}}^{n}}{\Delta t}
+R^n\nonumber,\]
\[\tag{624}
R^n = \frac{1}{2}{u_{\small\mbox{e}}}''(t_n)\Delta t + {\cal O}(\Delta t^2)\]
\[{u_{\small\mbox{e}}}'(t_{n+\theta}) =
[\bar D_t{u_{\small\mbox{e}}}]^{n+\theta} +R^{n+\theta} = \frac{{u_{\small\mbox{e}}}^{n+1} - {u_{\small\mbox{e}}}^{n}}{\Delta t}
+R^{n+\theta}\nonumber,\]
\[R^{n+\theta} = -\frac{1}{2}(1-2\theta){u_{\small\mbox{e}}}''(t_{n+\theta})\Delta t +
\frac{1}{6}((1 - \theta)^3 - \theta^3){u_{\small\mbox{e}}}'''(t_{n+\theta})\Delta t^2 +
\nonumber\]
\[\tag{625}
\quad {\cal O}(\Delta t^3)\]
\[{u_{\small\mbox{e}}}'(t_n) =
[D_t^{2-}{u_{\small\mbox{e}}}]^n +R^n = \frac{3{u_{\small\mbox{e}}}^{n} - 4{u_{\small\mbox{e}}}^{n-1} + {u_{\small\mbox{e}}}^{n-2}}{2\Delta t}
+R^n\nonumber,\]
\[\tag{626}
R^n = \frac{1}{3}{u_{\small\mbox{e}}}'''(t_n)\Delta t^2 + {\cal O}(\Delta t^3)\]
\[{u_{\small\mbox{e}}}''(t_n) =
[D_tD_t {u_{\small\mbox{e}}}]^n +R^n = \frac{{u_{\small\mbox{e}}}^{n+1} - 2{u_{\small\mbox{e}}}^{n} + {u_{\small\mbox{e}}}^{n-1}}{\Delta t^2}
+R^n\nonumber,\]
\[\tag{627}
R^n = -\frac{1}{12}{u_{\small\mbox{e}}}''''(t_n)\Delta t^2 + {\cal O}(\Delta t^4)\]
\[{u_{\small\mbox{e}}}(t_{n+\theta}) = [\overline{{u_{\small\mbox{e}}}}^{t,\theta}]^{n+\theta} +R^{n+\theta}
= \theta {u_{\small\mbox{e}}}^{n+1} + (1-\theta){u_{\small\mbox{e}}}^n +R^{n+\theta},\nonumber\]
\[\tag{628}
R^{n+\theta} = -\frac{1}{2}{u_{\small\mbox{e}}}''(t_{n+\theta})\Delta t^2\theta (1-\theta) +
{\cal O}(\Delta t^3)
{\thinspace .}\]
Finite differences of exponential functions¶
Complex exponentials¶
Let \(u^n = \exp{(i\omega n\Delta t)} = e^{i\omega t_n}\).
\[\tag{629}
[D_tD_t u]^n = u^n \frac{2}{\Delta t}(\cos \omega\Delta t - 1) =
-\frac{4}{\Delta t}\sin^2\left(\frac{\omega\Delta t}{2}\right),\]
\[\tag{630}
[D_t^+ u]^n = u^n\frac{1}{\Delta t}(\exp{(i\omega\Delta t)} - 1),\]
\[\tag{631}
[D_t^- u]^n = u^n\frac{1}{\Delta t}(1 - \exp{(-i\omega\Delta t)}),\]
\[\tag{632}
[D_t u]^n = u^n\frac{2}{\Delta t}i\sin{\left(\frac{\omega\Delta t}{2}\right)},\]
\[ \begin{align}\begin{aligned}\tag{633}
[D_{2t} u]^n = u^n\frac{1}{\Delta t}i\sin{(\omega\Delta t)}\\ {\thinspace .}\end{aligned}\end{align} \]
Real exponentials¶
Let \(u^n = \exp{(\omega n\Delta t)} = e^{\omega t_n}\).
\[\tag{634}
[D_tD_t u]^n = u^n \frac{2}{\Delta t}(\cos \omega\Delta t - 1) =
-\frac{4}{\Delta t}\sin^2\left(\frac{\omega\Delta t}{2}\right),\]
\[\tag{635}
[D_t^+ u]^n = u^n\frac{1}{\Delta t}(\exp{(i\omega\Delta t)} - 1),\]
\[\tag{636}
[D_t^- u]^n = u^n\frac{1}{\Delta t}(1 - \exp{(-i\omega\Delta t)}),\]
\[\tag{637}
[D_t u]^n = u^n\frac{2}{\Delta t}i\sin{\left(\frac{\omega\Delta t}{2}\right)},\]
\[ \begin{align}\begin{aligned}\tag{638}
[D_{2t} u]^n = u^n\frac{1}{\Delta t}i\sin{(\omega\Delta t)}\\ {\thinspace .}\end{aligned}\end{align} \]
Finite differences of \(t^n\)¶
The following results are useful when checking if a polynomial term in a solution fulfills the discrete equation for the numerical method.
\[\tag{639}
\lbrack D_t^+ t\rbrack^n = 1,\]
\[\tag{640}
\lbrack D_t^- t\rbrack^n = 1,\]
\[\tag{641}
\lbrack D_t t\rbrack^n = 1,\]
\[\tag{642}
\lbrack D_{2t} t\rbrack^n = 1,\]
\[ \begin{align}\begin{aligned}\tag{643}
\lbrack D_{t}D_t t\rbrack^n = 0\\ {\thinspace .}\end{aligned}\end{align} \]
The next formulas concern the action of difference operators on a \(t^2\) term.
\[\tag{644}
\lbrack D_t^+ t^2\rbrack^n = (2n+1)\Delta t,\]
\[\tag{645}
\lbrack D_t^- t^2\rbrack^n = (2n-1)\Delta t,\]
\[\tag{646}
\lbrack D_t t^2\rbrack^n = 2n\Delta t,\]
\[\tag{647}
\lbrack D_{2t} t^2\rbrack^n = 2n\Delta t,\]
\[\tag{648}
\lbrack D_{t}D_t t^2\rbrack^n = 2,\]
Finally, we present formulas for a \(t^3\) term:
\[\tag{649}
\lbrack D_t^+ t^3\rbrack^n = 3(n\Delta t)^2 + 3n\Delta t^2 + \Delta t^2,\]
\[\tag{650}
\lbrack D_t^- t^3\rbrack^n = 3(n\Delta t)^2 - 3n\Delta t^2 + \Delta t^2,\]
\[\tag{651}
\lbrack D_t t^3\rbrack^n = 3(n\Delta t)^2 + \frac{1}{4}\Delta t^2,\]
\[\tag{652}
\lbrack D_{2t} t^3\rbrack^n = 3(n\Delta t)^2 + \Delta t^2,\]
\[\tag{653}
\lbrack D_{t}D_t t^3\rbrack^n = 6n\Delta t,\]
Application of finite difference operators to polynomials and exponential
functions, resulting in the formulas above, can easily be computed by
some sympy code (from the file lib.py):
from sympy import *
t, dt, n, w = symbols('t dt n w', real=True)
# Finite difference operators
def D_t_forward(u):
return (u(t + dt) - u(t))/dt
def D_t_backward(u):
return (u(t) - u(t-dt))/dt
def D_t_centered(u):
return (u(t + dt/2) - u(t-dt/2))/dt
def D_2t_centered(u):
return (u(t + dt) - u(t-dt))/(2*dt)
def D_t_D_t(u):
return (u(t + dt) - 2*u(t) + u(t-dt))/(dt**2)
op_list = [D_t_forward, D_t_backward,
D_t_centered, D_2t_centered, D_t_D_t]
def ft1(t):
return t
def ft2(t):
return t**2
def ft3(t):
return t**3
def f_expiwt(t):
return exp(I*w*t)
def f_expwt(t):
return exp(w*t)
func_list = [ft1, ft2, ft3, f_expiwt, f_expwt]
To see the results, one can now make a simple loop over the different type of functions and the various operators associated with them:
for func in func_list:
for op in op_list:
f = func
e = op(f)
e = simplify(expand(e))
print e
if func in [f_expiwt, f_expwt]:
e = e/f(t)
e = e.subs(t, n*dt)
print expand(e)
print factor(simplify(expand(e)))
