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There are at least three ways in which the stability of a scheme can be tested. These are: 1) the direct method, 2) the energy method, and 3) von Neumann's method. 84). Note that u j is a weighted mean of u j and u j – 1 . 80)], we may write n+1 uj n n ≤ uj ( 1 – µ ) + uj – 1 µ . 89) 30 n+1 max ( j ) u j Basic Concepts n ≤ max ( j ) u j , provided that 0 ≤ µ ≤ 1. 90) n We have shown that for 0 ≤ µ ≤ 1 the solution u j remains bounded for all time. Therefore, 0 ≤ µ ≤ 1 is a sufficient condition for stability.

Now we define truncation error and accuracy for a finite-difference scheme. Here we define a finite-difference scheme as a finite-difference equation which approximates, term-by-term, a differential equation. It is easy to find an approximation to each term of a differential equation, and we have already seen that the error of such an approximation can be made as small as desired, almost effortlessly. This is not our goal, however. Our goal is to find an approximation to the solution of the differential equation.

4! 2 3 4 2 3 4  ( m∆t ) ( m∆t ) ( m∆t ) – q – ( m∆t )q′ + -----------------q′′ – -----------------q′′′ + -----------------q′′′′ – …  2! 3! 4!  2 3 ∆t ∆t = β f + ∆tf′ + -------- f′′ + -------- f′′′ + … 2! 3! + αn f + αn – 1 2 3 ∆t ∆t f – ∆tf′ + -------- f′′ – -------- f′′′ + … 2! 3! 2 3 2 3 ( 2∆t ) ( 2∆t ) + α n – 2 f – 2∆tf′ + ---------------- f′′ – ---------------- f′′′ + … 2! 3! ( 3∆t ) ( 3∆t ) + α n – 3 f – 3∆tf′ + ---------------- f′′ – ---------------- f′′′ + … 2! 3! +… 2 3 ( l∆t ) ( l∆t ) + α n – l f – l∆tf′ + --------------- f′′ – --------------- f′′′ + … 2!

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