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

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Outline1 ReviewSingle Step MethodsStabilityIVP StabilityAmplification factor2 Local Truncation Error & Consistency3 Global truncation error & Convergence() Truncation Error, Consistency, and Convergence 8 October 2012 2 / 54
Single-step methodsSummaryWe have 4 schemes to solvey ′ (t) = f (t, y(t))y(t 0 ) = y 0 .Forward Euler: w n+1 = w n + hf (t n , w n )Backward Euler: w n+1 = w n + hf (t n+1 , w n+1 )Trapezoidal method:Modified Euler:w n+1 = w n + h 2 (f (t n, w n ) + f (t n+1 , w n+1 ))¯w n+1 = w n + hf (t n , w n )w n+1 = w n + h 2 (f (t n, w n ) + f (t n+1 , ¯w n+1 ))() Truncation Error, Consistency, and Convergence 8 October 2012 3 / 54
Page 1: Lecture 6:Truncation Error, Consist
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 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
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Lecture 6: =1=Truncation Error, Consistency, and Convergence

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