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

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Forward EulerEuler forward,w n+1 = w n + hF (t n , w n ),Is an explicit method.Has amplification factor Q(hλ) = 1 + hλ.Is stable for 0 < h < 2|λ| .Has local truncation error of order one.Stability + Consistency = Convergence. It is a first ordermethod.() Truncation Error, Consistency, and Convergence 8 October 2012 18 / 54
ConsistencyLet us do the same for the other methods.Convergence is difficult to prove.Lax theorem says that we can instead prove stability andconsistency.Stability is given by ensuring the amplification factor is lessthan one.|Q(hλ)| ≤ 1.For consistency, we need the local truncation error tovanish as the step size becomes smaller.lim τ 1n+1(h) = limh→0 h→0 h (y n+1 − z n+1 ).() Truncation Error, Consistency, and Convergence 8 October 2012 19 / 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 and 10: Local Truncation ErrorLast time we
Page 11 and 12: Local Truncation ErrorFor consisten
Page 13 and 14: ConsistencyDefinition (Consistent)A
Page 15 and 16: Local truncation errorThe numerical
Page 17: Local truncation errorForward Euler
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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