2
$\begingroup$

When reviewing papers on the numerical solutions of Partial Differential Equations (PDEs), I observed the following equation: $$ (1-C\tau)||\theta^n||^2 \leq ||\theta^{n-1}||^2 + C\tau (h^{2r+2}), $$ where $\tau$ is the discrete time step, and $h$ is the spactial step size. The above equation is equal to $$ ||\theta^n||^2 \leq (1+C\tau)||\theta^{n-1}||^2 + C\tau (h^{2r+2}), $$ What is the reason why the above formulas are equivalent?

$\endgroup$
2
  • 1
    $\begingroup$ $C$ is typically considered a "generic constant" in these sorts of estimates, i.e., it may have a different value in the two equations. See if you can show the equivalence if (i) you rename $C$ to $D$ in the second line, and (ii) ignore all terms higher order in $\tau$ and $h$. $\endgroup$ Dec 20, 2023 at 4:37
  • $\begingroup$ @WolfgangBangerth Thx! ignoring all terms of higher order is a good idea $\endgroup$
    – Owen Jun
    Dec 20, 2023 at 8:02

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.