20 - Numerical Differentiation IVPs for ODEs Discretisation...

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Numerical Differentiation IVPs for ODEs Discretisation & one-step methods Initial-Value Problems for Ordinary Differential Equations Dhavide Aruliah UOIT MATH 2070U c ± D. Aruliah (UOIT) MATH 2070U 1 / 32

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Numerical Differentiation IVPs for ODEs Discretisation & one-step methods Initial-Value Problems for Ordinary Differential Equations 1 Numerical Differentiation 2 Initial-value problems for ordinary differential equations Differential equations and initial-value problems Solutions of DEs and IVPS 3 Discretisation of differential equations: one-step methods Euler’s method Accuracy and Stability Modiﬁcations of Euler’s method c ± D. Aruliah (UOIT) MATH 2070U 2 / 32
Numerical Differentiation IVPs for ODEs Discretisation & one-step methods Implementation in M ATLAB with diff diff computes differences of successive elements in a vector For nonuniform grids, use in conjunction with vectorised operator ./ for convenience e.g., following implements forward differences ( f ( x k + 1 ) - f ( x k ) / ( x k + 1 - x k ) on nonuniform grid x = [1,1.25,1.3,1.5,1.7,1.9,2.0]; % nonuniform grid in x y1 = 5*x-3; % Compute first function on grid x y2 = 3*x.^2-4*x+2; % Compute second function on grid x dy1dx_f = diff(y1)./diff(x); % Forward differences dy2dx_f = diff(y2)./diff(x); % Forward differences c ± D. Aruliah (UOIT) MATH 2070U 4 / 32

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Numerical Differentiation IVPs for ODEs Discretisation & one-step methods Example A car laps a race track in 84 s. The speed of the car (in ft/s) is determined at 6 second intervals starting from the beginning of the lap using a radar gun. time 0 6 12 18 24 30 36 42 speed 124 134 148 156 147 133 121 109 time 48 54 60 66 72 78 84 speed 99 85 78 89 104 116 123 What is the acceleration of the car at any given time? Let v ( t ) denote speed at time t , so acceleration is v 0 ( t ) Here, h = 6 s is time-step between measurements Use v 0 ( t ) ( δ v )( t ) = ( v ( t + h ) - v ( t - h )) / ( 2 h ) for t = 6, 12, . . . , 78 Use v 0 ( 0 ) ( δ v )( 0 ) = ( v ( h ) - v ( 0 )) / h at t = 0 Use v 0 ( 84 ) ( δ v )( 84 ) = ( v ( 84 ) - v ( 84 - h )) / h at t = 84 c ± D. Aruliah (UOIT) MATH 2070U 5 / 32
Numerical Differentiation IVPs for ODEs Discretisation & one-step methods Example (cont.) v 0 ( 0 ) v ( 6 ) - v ( 0 ) 6 v 0 ( 6 ) v ( 12 ) - v ( 0 ) 2 · 6 ··· v 0 ( 84 ) v ( 84 ) - v ( 78 ) 6 = 134 - 124 6 = 148 - 124 12 = 123 - 116 6 = 1.666667 = 2.000000 = 1.166667 t 0 6 12 18 24 30 36 42 v ( t ) 124 134 148 156 147 133 121 109 v 0 ( t ) 1.667 2.00 1.833 - 0.0833 - 1.917 - 2.167 - 2.00 - 1.833 t 48 54 60 66 72 78 84 v ( t ) 99 85 78 89 104 116 123 v 0 ( t ) - 2.00 - 1.75 0.3333 2.167 2.25 1.583 1.167 v=[124,134,148,156,147,133,121,109,99,85,78,89,104,116,123]; h=6; % Size of time-step/mesh-spacing vprime=gradient(v,h) % Need h to set scale properly c ± D. Aruliah (UOIT) MATH 2070U 6 / 32

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Numerical Differentiation IVPs for ODEs Discretisation & one-step methods M ATLAB routines for differentiation diff Differences of adjacent elements of vector gradient Gradient of scalar ﬁeld using differences del2 Laplacian of scalar ﬁeld divergence Divergence of vector ﬁeld curl Curl of vector ﬁeld polyder Analytic derivative of polynomial diff differences columns of 2D arrays MATLAB gradient command computes gradient ﬁeld Numerical differences used to approximate derivatives Centred differences in interior of 2D array Forward/backward differences at boundary of 2D array c ± D. Aruliah (UOIT) MATH 2070U 7 / 32
Numerical Differentiation IVPs for ODEs Discretisation & one-step methods Using M ATLAB ’s gradient hx = 0.2; hy = 0.1; % Spacing in x & y directions

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20 - Numerical Differentiation IVPs for ODEs Discretisation...

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