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?

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    $\begingroup$ $C$ 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


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