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Note accuracy determines formally how fast the errors tend to zero as the time step d t tends to zero.
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Stability determines if the errors build up or reinforce each other as one iterates in time
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All time differencing schemes are valid representations of equations i.e.
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reduce to differential equation in limit of dx and d t tend to zero
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However not all differencing schemes are stable!
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We can investigate stability for simple linear equations with von Neumann's Analysis
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Let u(n)j be replaced by u(n)j + e(n)j where:
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n labels time iteration and j spatial discretization
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If the equations are linear in u, the e(n)j obeys same equations as u(n)j but with simple (zero) boundary conditions
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If equation not linear, then this analysis still governs local behavior and gives a stability condition must must be satisfied by linearized equations
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To linearize something like u ¶ (u) / ¶ t, replace varying u outside derivative by its local value.
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