Lecture 6: =1=Truncation Error, Consistency, and Convergence

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StabilityLast time, we covered stability.Stability is given by the amplification factor,w n+1 = Q(hλ)w n .For this, we choose h so that |Q(hλ)| ≤ 1.() Truncation Error, Consistency, and Convergence 8 October 2012 10 / 54
Local Truncation ErrorFor consistency, we will need the local truncation error. Butfirst we need a few definitions:y n+1 = exact solution at time t n+1 .z n+1 = Q(hλ)y n = approximation applied to the exactsolution at time t n .w n+1 = (Q(hλ)) n+1 y 0 = approximation applied n + 1times.() Truncation Error, Consistency, and Convergence 8 October 2012 11 / 54
Page 1 and 2: Lecture 6:Truncation Error, Consist
Page 3 and 4: Single-step methodsSummaryWe have 4
Page 5 and 6: StabilityIVP StabilityFor the stabi
Page 7 and 8: StabilityNumerical StabilityFor num
Page 9: Local Truncation ErrorLast time we
Page 13 and 14: ConsistencyDefinition (Consistent)A
Page 15 and 16: Local truncation errorThe numerical
Page 17 and 18: Local truncation errorForward Euler
Page 19 and 20: ConsistencyLet us do the same for t
Page 21 and 22: Modified EulerModified Euler is giv
Page 23 and 24: Local truncation errorBackward Eule
Page 25 and 26: Backward EulerEuler backward,w n+1
Page 27 and 28: Local truncation errorTrapezoidal m
Page 29 and 30: Local truncation errorTo find the l
Page 31 and 32: Local truncation errorFor explicit
Page 33 and 34: Local truncation errorFinding Φ(z
Page 35 and 36: Global truncation errorFor that we
Page 37 and 38: ConvergenceWe now have the three el
Page 39 and 40: Higher Order Methods4 th −order R
Page 41 and 42: ExampleExampleConsider the numerica
Page 43 and 44: ExampleTruncation errorSubstituting
Page 45 and 46: ExampleTruncation errorWhich gives
Page 47 and 48: ExampleAmplification factorSimplify
Page 49 and 50: ExampleNonlinear equationThe linear
Page 51 and 52: ExampleNonlinear equation|Q(hλ)| =
Page 53 and 54: SummaryLax Equivalence Theorem: For

Lecture 6: =1=Truncation Error, Consistency, and Convergence

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