Introduction to standard and non-standard Numerical Methods

Transcription

1 Introduction to standard and non-standard Numerical Methods Dr. Mountaga LAM AMS : African Mathematic School 2018 May 23, 2018

2 One-step methods Runge-Kutta Methods Nonstandard Finite Difference Scheme Application of Polio model

3 Solving dx dt = f (x) Are numerical methods whose to forward a step, only the previous step information is needed, ie step n+1 only depends on the step n. Euler s idea

4 Solving dx dt = f (x) Are numerical methods whose to forward a step, only the previous step information is needed, ie step n+1 only depends on the step n. Euler s idea I can t solve the equation because I don t know what dx is. So dt pick a small number h > 0 and say that dx dt x(t + h) x(t) h

5 Solving dx dt = f (x) Are numerical methods whose to forward a step, only the previous step information is needed, ie step n+1 only depends on the step n. Euler s idea I can t solve the equation because I don t know what dx is. So dt pick a small number h > 0 and say that dx dt The differential equation then becomes x(t + h) x(t) h x(t + h) x(t) h f (x)

6 Solving dx dt = f (x) Are numerical methods whose to forward a step, only the previous step information is needed, ie step n+1 only depends on the step n. Euler s idea I can t solve the equation because I don t know what dx is. So dt pick a small number h > 0 and say that dx dt The differential equation then becomes x(t + h) x(t) h x(t + h) x(t) h f (x) If you know x(t) and h then you can solve this equation for x(t + h).

7 Solving dx dt = f (x) has solution x(t + h) x(t) h f (x) x(t + h) x(t) + h f (x)

8 Solving dx dt = f (x) has solution x(t + h) x(t) h f (x) x(t + h) x(t) + h f (x) Exemple (t = 0): If we know x(0), then this equation allows us to compute x(0 + h) = x(h). Exemple (t = h): If we know x(h), then this equation allows us to compute x(h + h) = x(2h). and than x(2h + h) = x(3h), x(3h + h) = x(4h)...

9 Solving dx dt = f (x) Pick small number of h and compute x(h) = x(0) + h f (0) x(2h) = x(h) + h f (h) x(3h) = x(2h) + h f (2h) x(4h) = x(3h) + h f (3h). How did Euler do this? R: By hand (main...), if yes! How many times? How do we do this in the 21st century? With computer

10 Idea is that New value = Old value+ step size*slope x n+1 = x n + h nf (t n, x n) Slope is generally a function of t, hence x(t) Different methods differ in how to estimate φ x f (x) = 0.2 x 2 x(t i+1 ) x i+1 error x i t i h t i+1 t

11 Euler s Method (RK method with order 1) A one-step method expresses x n+1 in terms of the previous value x n. The simplest example of a one-step method for the numerical solution of the initial value problem (IVP) is Euler s method. Given that x(t 0 ) = x 0 We define x n+1 = x n + h nf (t n, x n)

12 The midpoint rule (RK method with order two) x n+1 = x n + hf (t n + h 2, xn + h f (xn, tn)) 2

13 Second-Order Runge-Kutta Methods Let x (t) = f (t, x(t)). where x(t + h) = x(t) + hx (t) + h2 2! x (t) + O(h 3 ) x (t) = f t(t, x) + f x (t, x)x (t) = f t(t, x) + f x (t, x)f (t, x) with Jacobian f x x(t + h) = x(t) + hf (t, x) + h2 [ ] f t(t, x) + f x (t, x)f (t, x) + O(h 3 ) 2!

14 Second-Order Runge-Kutta Methods x(t + h) = x(t) + h 2 f (t, x) + h 2 [ ] f t(t, x) + hf t(t, x) + hf x (t, x)f (t, x) + O(h 3 ) Recalling the multivariate Taylor expansion f (t + h, x + k) = f (t, x) + hf t(t, x) + f x (t, x)k + Then x(t + h) = x(t) + h 2 f (t, x) + h 2 [ ] f (t + h, x + hf (t, x)) + O(h 3 ) we get the numerical method : 1 x n+1 = x n + h( 2 k ) 2 k 2 with k 1 = f (t n, x n) and k 2 = f (t n + h, x n + hk 1 )

15 Fourth-Order Runge-Kutta Methods The classical method is given by x n+1 = x n + h ) (k 1 + 2k 2 + 2k 3 + k 4 6 with k 1 is the increment based on the slope at the beginning of the interval, using x (Euler s method); k 2 is the increment based on the slope at the midpoint of the interval, using x and k 1 k 3 is again the increment based on the slope at the midpoint, but now using x and k 2 k 4 is the increment based on the slope at the end of the interval, using x and k 3

16 Butcher tableau The family of explicit Runge-Kutta methods is a generalization of the RK4 method mentioned above. It is given by where s x n+1 = x n + h b i k i i=1 k 1 = f (t n + 0h, x n), k 2 = f (t n + c 2 h, x n + h(a 21 k 1 )), k 3 = f (t n + c 3 h, x n + h(a 31 k 1 + a 32 k 2 )), k 4 = f (t n + c 4 h, x n + h(a 41 k 1 + a 42 k 2 + a 43 k 3 )),. k s = f (t n + c sh, x n + h(a s1 k 1 + a s1 k a s,s 1 k s 1 )), And the final step to combine these intermediate steps like this: x n+1 = x n + b 1 k 1 + b 2 k 2 + b 3 k 3 + b 4 k b sk s The Runge-Kutta method is consistent if i 1 a ij = c i, for i = 2,, s. j=1

17 Exemple : The explitcit RK4 method falls in this framework It is not difficult to construct s-stage implicit methods which are A-stable. For example, this can be done by choosing the coefficients c i and b i to be the quadrature points and weights respectively in the Gauss quadrature formula for the evaluation of 1 s f (t)dt b i f (c i ) 0 i=1 The numbers a ij can then be chosen so that the method has order 2s, and is A-stable.

18 Let us observe that on expanding x(t n+1 ) = x(t n + h) into a Taylor series x (t n) = f (t n, x(t n)) we have that : x(t n+1 ) = x(t n) + hf (t n, x(t n)) + O(h 2 )

19 More generally, a one-step method may be written in the form x(t n+1 ) = x(t n) + hφ(t n, x n, h), n = 0, 1,, N 1, x(t 0 ) = x 0, where we assume that φ : [t 0, t 0 + T ] R R R is a continuous function. In pratical case, the function φ(t, x, h) can be define For example : Euler s Method is given by φ(t n, x n, h) = f (t n, x n)

20 Global error In order to assess the accuracy of the numerical method, we define the global error, e n, by The truncation error, T n, is define by e n = x(t n) x n T n = x(t n+1) x(t n) h φ(t n, x(t n); h).

21 Convergence Consider the general one-step method where, in addition to being a continuous function of its arguments, φ is assumed to satisfy a Lipschitz condition with respect to its second argument, that is, there exists a positive constant L φ such that, for 0 h h 0 and for (t, u) and (t, v) in the rectangle D = {(t, x) : t 0 x t M, x x 0 C} we have that φ(t, u; h) φ(t, v; h) L φ u v. Then, assuming that x x 0 C, n = 1, 2,, N it follows that where T = max 0 n N 1 T n e n T ) (e L φ(t n t 0 ) 1 L φ

22 Consistance Let us apply this general result in order to obtain a bound on the global error in Euler s method. The truncation error for Euler s method is given by Taylor series : T n = x(t n+1) x(t n) h f (t n, x(t n)) = x(t n+1) x(t n) h x (t n) x(t n+1 ) = x(t n) + hx (t n) + h2 2! x (ζn), t n < ζ n < t n+1. T n = 1 2 hx (ζ n) M = max x ζ [t 0,t M ] T n T = 1 [ e L(t n t 0 ) 2 M 1 ], n = 0, 1,, N. L Remark : In practice, for such h we shall have x(t n) x n = e n Tol for any n = 0, 1,, N. So h (expression Tol)

23 Bref definition A Scheme is said explicit if x(t i+1 ) can be write as a linear combination of x(t i ), f (x k, t i1 ),... k. The scheme is implicite if it others values are necessar.

24 Bref definition A Scheme is said explicit if x(t i+1 ) can be write as a linear combination of x(t i ), f (x k, t i1 ),... k. The scheme is implicite if it others values are necessar. The scheme is said one-step scheme if we use only two values of times (i.e. t i and t i+1 ), otherwise it will be a multi step scheme.

25 Bref definition A Scheme is said explicit if x(t i+1 ) can be write as a linear combination of x(t i ), f (x k, t i1 ),... k. The scheme is implicite if it others values are necessar. The scheme is said one-step scheme if we use only two values of times (i.e. t i and t i+1 ), otherwise it will be a multi step scheme. The convergence means that the numerical solution of the numerical scheme tend to the solution of the ordinary differentiale equation.

26 Bref definition A Scheme is said explicit if x(t i+1 ) can be write as a linear combination of x(t i ), f (x k, t i1 ),... k. The scheme is implicite if it others values are necessar. The scheme is said one-step scheme if we use only two values of times (i.e. t i and t i+1 ), otherwise it will be a multi step scheme. The convergence means that the numerical solution of the numerical scheme tend to the solution of the ordinary differentiale equation. The discretisation scheme M h, t of operator L is consistant if the function φ is smooth enought and lim h, t 0 (Mφ M h, tφ) = 0

27 Bref definition A Scheme is said explicit if x(t i+1 ) can be write as a linear combination of x(t i ), f (x k, t i1 ),... k. The scheme is implicite if it others values are necessar. The scheme is said one-step scheme if we use only two values of times (i.e. t i and t i+1 ), otherwise it will be a multi step scheme. The convergence means that the numerical solution of the numerical scheme tend to the solution of the ordinary differentiale equation. The discretisation scheme M h, t of operator L is consistant if the function φ is smooth enought and lim h, t 0 (Mφ M h, tφ) = 0 A linear scheme consistant is convergent if only if it is stable.

28 Nonstandard Finite Difference Scheme (NFDS) Exact schemes : Examples Let dx dt = f (t, x) = λx and x(t 0) = x 0 The general solution is given by : then the exact scheme is x(t) = x 0 e λ(t t 0). u k+1 = e λh u k.

29 NFDS : Examples Let dx dt = f (t, x) = λx and x(t 0) = x 0 x k+1 = e λh x k. Combining the previous exact schemes, the NFDS scheme is given by : ( 1 e x k+1 x k = (e λh λh ) 1)x k = λ x k, (1) λ than x k+1 x k φ(h) = λx k avec φ(h) = 1 e λh λ (2)

30 The Lotka-Volterra system: Euler s Method with { xk+1 x k φ 1 (h) = ax k bx k+1 y k, y k+1 y k φ 2 (h) = cy k+1 + dx k+1 y k, φ 1 (h) = 1 e λh λ and φ 2 (h) = 1 e λh λ Figure: Phase plane and time history : Euler explicit method and NFDS method of Lotka-Volterra system (a=1;b=0.01;c=1;d=0.02)

31 The Lotka-Volterra system: Euler s Method with { xk+1 x k φ 1 (h) = ax k bx k+1 y k, y k+1 y k φ 2 (h) = cy k+1 + dx k+1 y k, φ 1 (h) = 1 e λh λ and φ 2 (h) = 1 e λh λ Figure: Phase plane and time history : Euler explicit method and NFDS method of Lotka-Volterra system (a=1;b=0.01;c=1;d=0.02)

32 Mikens Rules Pr. Ronald Mickens (Atlanta, USA) used the construction of exact schemes to propose simple rules to develop nonstandard finite difference schemes for differential equations and even partial differential equations. Rule 1 The orders of the discrete derivatives should be equal to the orders of the corresponding derivatives of the differential equation.

33 Mikens Rules Pr. Ronald Mickens (Atlanta, USA) used the construction of exact schemes to propose simple rules to develop nonstandard finite difference schemes for differential equations and even partial differential equations. Rule 1 The orders of the discrete derivatives should be equal to the orders of the corresponding derivatives of the differential equation. Rule 2 Denominator functions for the discrete derivatives must, in general, be expressed in terms of more complicated functions of the step-size than those conventionally used

34 Mikens Rules Pr. Ronald Mickens (Atlanta, USA) used the construction of exact schemes to propose simple rules to develop nonstandard finite difference schemes for differential equations and even partial differential equations. Rule 1 The orders of the discrete derivatives should be equal to the orders of the corresponding derivatives of the differential equation. Rule 2 Denominator functions for the discrete derivatives must, in general, be expressed in terms of more complicated functions of the step-size than those conventionally used Rule 3 Nonlinear terms should be, in general, be replaced by nonlocal discrete representations.

35 Mikens Rules Pr. Ronald Mickens (Atlanta, USA) used the construction of exact schemes to propose simple rules to develop nonstandard finite difference schemes for differential equations and even partial differential equations. Rule 1 The orders of the discrete derivatives should be equal to the orders of the corresponding derivatives of the differential equation. Rule 2 Denominator functions for the discrete derivatives must, in general, be expressed in terms of more complicated functions of the step-size than those conventionally used Rule 3 Nonlinear terms should be, in general, be replaced by nonlocal discrete representations. Rule 4 Special conditions that hold for the solutions of the differential equations should also hold for the solutions of the finite difference scheme.

36 Mikens Rules Pr. Ronald Mickens (Atlanta, USA) used the construction of exact schemes to propose simple rules to develop nonstandard finite difference schemes for differential equations and even partial differential equations. Rule 1 The orders of the discrete derivatives should be equal to the orders of the corresponding derivatives of the differential equation. Rule 2 Denominator functions for the discrete derivatives must, in general, be expressed in terms of more complicated functions of the step-size than those conventionally used Rule 3 Nonlinear terms should be, in general, be replaced by nonlocal discrete representations. Rule 4 Special conditions that hold for the solutions of the differential equations should also hold for the solutions of the finite difference scheme. Rule 5 The scheme should not introduce extraneous or spurious solutions.

37 Mikens Rules Pr. Ronald Mickens (Atlanta, USA) used the construction of exact schemes to propose simple rules to develop nonstandard finite difference schemes for differential equations and even partial differential equations. Rule 1 The orders of the discrete derivatives should be equal to the orders of the corresponding derivatives of the differential equation. Rule 2 Denominator functions for the discrete derivatives must, in general, be expressed in terms of more complicated functions of the step-size than those conventionally used Rule 3 Nonlinear terms should be, in general, be replaced by nonlocal discrete representations. Rule 4 Special conditions that hold for the solutions of the differential equations should also hold for the solutions of the finite difference scheme. Rule 5 The scheme should not introduce extraneous or spurious solutions. Rule 6 For N differential system, it could be useful to construct nonstandard schemes for subsystems of M < N differential equations and to combine them to obtain a consistant scheme.

38 Some examples of NFDS of ODE Équations NFDS Denominators dx dt = λx dx dt = λx dx dt = x2 dx dt = λ 1x λ 2 x 2 dx dt = λ 1x λ 2 x 2 d 2 x dt 2 + ω 2 x = 0 d 2 x dt 2 = λ dx dt x k+1 x k = λx φ(h) k φ(h) = 1 e λh λ = λx φ(h) k+1 φ(h) = eλh 1 λ = x φ(h) k x k+1 φ(h) = h x k+1 x k x k+1 x k x k+1 x k = λ φ(h) 1 x k λ 2 x k x k+1 φ(h) = eλ 1 h 1 λ 1 = λ φ(h) 1 x k+1 λ 2 x k x k+1 φ(h) = 1 e λ 1 h λ 1 ( ) + ω 2 x k = 0 φ(h) = 2 ω sin hω 2 x k+1 x k x k+1 2x k +x k 1 φ 2 (h) x k+1 2x k +x k 1 φ 1 (h) = λ x k+1 x k φ 2 (h) φ 1 (h) = ( eλh 1 )h; φ λ 2 (h) = h

39 Fundamental Rules? it is fundamental to follow rules 2 and 3. The others rules then follow... to build the best numerical scheme we need a deep theoretical study of the continuous problem is helpful to capture the properties of the solution and the problem

40 Re-formulation for applying GAS in dynamical systems: Kamgang-Sallet where dx dt = A(x)x + f, x(0) = x 0, ds = (1 π)ρ (τ + ϑ + µ)s, dt dv dt = πρ + τs (µ + γ)v, di dt = ϑs (α + δ + µ + κ)i, dj dt = δi µj, dg dt = ξi ηg, dr dt = γv + αi µr, I ϑ = β h N + β G G, y = (S, V, R), z = (I, J, G) G + K { dy = A dt 1 (x)(y y ) + A 12 (x)z, dz = A dt 2 (x)z,

41 Application of GAS with numerical dynamical systems : Kamgang and Sallet chose of φ(h) is given by { 1 φ(h) min µ + γ, 1 α + δ + µ + κ, 1 µ, 1 }. η Q such that Q max 1 2 { λ i }, i = 1,, 5, φ(h) = ϕ(qh) Q avec ϕ(z) = 1 e z, z R +.

42 Implicit scheme S k+1 S k φ(h) = I (1 π)ρ (τ + β k h N + β k G G k G k +K + µ)sk+1, V k+1 V k φ(h) = πρ + τs k+1 (µ + γ)v k+1, I k+1 I k I = (β k φ(h) h N + β k G G k G k +K )Sk+1 (α + δ + µ + κ)i k+1, J k+1 J k φ(h) = δi k+1 µj k+1, G k+1 G k φ(h) = ξi k+1 ηg k+1, R k+1 R k φ(h) = γv k+1 + αi k+1 µr k+1, y k+1 y k φ(h) = A 1 (x k )(y k+1 y ) + A 12 (x k )z k+1, z k+1 z k φ(h) = A 2 (x k )z k+1,

43 explicit scheme S k+1 S k φ(h) = I (1 π)ρ (τ + β k h N + β k G G k G k +K + µ)sk+1, V k+1 V k φ(h) = πρ + τs k (µ + γ)v k, I k+1 I k I = (β k φ(h) h N + β k G G k G k +K )Sk+1 (α + δ + µ + κ)i k, J k+1 J k φ(h) = δi k µj k, G k+1 G k φ(h) = ξi k ηg k, R k+1 R k φ(h) = γv k + αi k µr k, y k+1 y k φ(h) = A 1 (x k )(y k y ) + A 12 (x k )z k, z k+1 z k φ(h) = A 2 (x k )z k,

44 Simulations Figure: R 0 > 1.

45 Simulations Figure: R 0 < 1.

46 Comparison Figure: Comparison between NFDS and RK4

47 Comparison Figure: Comparison between NFDS and RK4