What is consistency?

by Yi Zhang

In numerical methods one often encounters the consistency. In spite of all the fuss, many textbooks fail to clearify the “idea” of consistency. In fact, even from the name, it is means of preventing comparing “wrong things”.

Numerical methods always dealing with approximating exact solutions, one kind or another. Even equipped with tools that essentially requir the error between approximate solution and exact solution, and even with small errors, one can still end up with totally wrong answers. It is like I have an apple, and try to immitate it with a pare, even somehow I can make the latter bear very similar shape, color, or size and density, as long as it tastes pear, it’s not an apple. Analogically, consistency is the way to ensure that when I have this fruit of same color, shape, size, etc., I can be sure it’s an apple. In finite difference, it is in the version of “when the local truncation error shrinks, the numerical solution goes towards the exact solution”.

As most of the concept in applied mathematics, consistency of order n for a numerical scheme can be defined/interpretated in various ways, here are some I know, and Taylor expansion is among the most frequently used tool to check consistency.

  1. \|Lu-L_hu\|=O(h^n)
  2. For any element p(x) of a basis of space of order n polynomial, the interpolation by set \{\Psi_i(x)\} reproduces p(x) to order n, i.e.
    \sum_i D^{\alpha}\Psi_i(x)p(x_i)=D^{\alpha}p(x),\quad |\alpha|\leq n
  3. Any exact solution with order less than or equal to n can be recovered by numerical schemeL_h
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