Truncation Error Analysis via sympy

Copyright (C) 2020 Andreas Kloeckner

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In [42]:
import sympy as s

Establish some variables that we'll need:

In [43]:
u = s.Function("u")
a, x, t, h_x, h_t = s.symbols("a, x, t, h_x, h_t")
xi_1, xi_2, tau = s.symbols("xi1, xi2, tau")

taylor is a utility function that spits out a taylor expansion for $f(x+h)$, optionally including a remainder term, with all variables under our control.

In [44]:
def taylor(f, x, h, n, remainder_variable=None):
    result = sum(f.diff(x, i)*h**i/s.factorial(i) for i in range(n))
    if remainder_variable:
        result += f.diff(x, n).subs(x, remainder_variable)*h**n/s.factorial(n)
    return result
  • Try it out by expanding $u(x+h_x,t)$
  • Vary the order
  • Expand $u(x,t+h_t)$ instead
In [48]:
taylor(u(x,t), x, h_x, 3, xi)
$\displaystyle \frac{h_{x}^{3} \frac{\partial^{3}}{\partial \xi^{3}} u{\left(\xi,t \right)}}{6} + \frac{h_{x}^{2} \frac{\partial^{2}}{\partial x^{2}} u{\left(x,t \right)}}{2} + h_{x} \frac{\partial}{\partial x} u{\left(x,t \right)} + u{\left(x,t \right)}$

Assign the PDE we're solving to pde:

In [49]:
pde = u(x, t).diff(t) + a * u(x, t).diff(x)
$\displaystyle a \frac{\partial}{\partial x} u{\left(x,t \right)} + \frac{\partial}{\partial t} u{\left(x,t \right)}$

Write out the scheme we're analyzing, in this case ETCS:

In [37]:
etcs = (
    (u(x, t+h_t) - u(x, t))/h_t
    a*(u(x+h_x, t) - u(x-h_x, t))/(2*h_x))
$\displaystyle \frac{a \left(- u{\left(- h_{x} + x,t \right)} + u{\left(h_{x} + x,t \right)}\right)}{2 h_{x}} + \frac{- u{\left(x,t \right)} + u{\left(x,h_{t} + t \right)}}{h_{t}}$

Follow this general pattern:

.subs(u(x, t+h_t), taylor(u(x,t), t, h_t, 2, tau))

to arrive at the truncation error.

⚠️ Make sure to keep the two $x$ remainder terms separate.

In [50]:
etcs_taylor = (
    .subs(u(x, t+h_t), taylor(u(x,t), t, h_t, 2, tau))
    .subs(u(x+h_x, t), taylor(u(x,t), x, h_x, 3, xi_1))
    .subs(u(x-h_x, t), taylor(u(x,t), x, -h_x, 3, xi_2))
sp.simplify(etcs_taylor - pde)
$\displaystyle \frac{a h_{x}^{2} \frac{\partial^{3}}{\partial \xi_{1}^{3}} u{\left(\xi_{1},t \right)}}{12} + \frac{a h_{x}^{2} \frac{\partial^{3}}{\partial \xi_{2}^{3}} u{\left(\xi_{2},t \right)}}{12} + \frac{h_{t} \frac{\partial^{2}}{\partial \tau^{2}} u{\left(x,\tau \right)}}{2}$
In [ ]: