Cole-Hopf transformation as numerical tool for the Burgers Equation. Alejandro Pozo

Transcription

1 Cole-Hopf transformation as numerical tool for the Burgers Equation A paper by Taku Ohwada Alejandro Pozo July 29 th, 2011

2 Outline 1 Cole-Hopf transformation 2 Numerical schemes 3 Some results

3 Outline 1 Cole-Hopf transformation 2 Numerical schemes 3 Some results

4 Cole-Hopf transformation The Burgers Equation u t + u u x = ν 2 u x 2 using the Cole-Hopf transformation, given by u(x, t) = 2ν x Θ(x, t) Θ(x, t) is transformed into the linear diffusion equation Θ t = ν 2 Θ x 2 that, for the initial value problem, has the solution Θ(x, t) = 1 2 πνt Θ(y, 0)e (y x)2 4νt dy

5 Cole-Hopf transformation Let us observe that u(x, t) = 2ν For convenience, we denote: x Θ(x, t) = Θ(x, t) = e 1 x 2ν u(ξ,t)dξ α Θ(x, t) G(x, y) := e (y x)2 4νt Then: u(x, t) = (y x)θ(y, 0)G(x, y)dy t Θ(y, 0)G(x, y)dy

6 Outline 1 Cole-Hopf transformation 2 Numerical schemes 3 Some results

7 Numerical schemes We have: Θ(x, t) = e 1 x 2ν u(ξ,t)dξ α u(x, t) = The logical computation would be: 1 Compute Θ(y, 0) using u(y, 0) 2 Compute u(x, t) using Θ(y, 0) (y x)θ(y, 0)G(x, y)dy t Θ(y, 0)G(x, y)dy However, the magnitude of Θ(y, 0) may become huge or vanishingly small, usual programming languages cannot deal with it. That is why the Cole-Hopf transformation may not seem really useful.

8 Numerical schemes We can express the constant α in Θ(y, 0) depending on x. Taking α = x, we can rewrite Θ(y, 0) as Θ(y, 0; x) := Θ(y, 0) = e 1 2ν y u(ξ,0)dξ x It satisfies: Θ(x, 0; x) = 1 u(y, 0) C 1 = Θ(y, 0; x) e C2 y x On the other hand G(x, y) decays double-exponentially as y x increases. Therefore, ΘG and (y x)θg are effectively zero for y x 1 and, if νt 1, we can compute u(x, t) using Θ(y, 0; x) in the neighborhood of y = x. Then, we will just use u(x, t) as the initial data for the following step.

9 Numerical schemes Now, we just need approximate the initial data u(ξ, 0) with respect to space. We will consider a uniform discretization x k = k x and a sufficiently small time-step t that satisfies ( x)2 4ν t 1, so that only the data of Θ(y, 0; x k) in [x k 1, x k+1 ] is necessary for the computation of u(x k, t). Some additional notation: η = ξ x k u k = u(x k, 0) We shall consider three different approaches. Using piecewise linear polynomials (u(x k + η, 0) = bη + a) Using cubic polynomials (u(x k + η, 0) = cν 3 + bη + a) Using quartic polynomials (u(x k + η, 0) = eν 4 + dν 3 + cν 2 + bν + a)

10 Numerical schemes Scheme A: Piecewise linear polynomials u(x k + η, 0) = ± u k±1 u k η + u k, 0 < ±η < x x Therefore, the exponent of ΘG becomes a piecewise quadratic polynomial of ν and, if t satisfies that 1 + b ± t > 1, the coefficient of η 2 is negative and the integration in the formula for u(x, t) can be done analytically. Even the integration on (, ) can be done safely.

11 Numerical schemes Scheme B: Piecewise cubic polynomials u(x k + η, 0) = u k+1 2u k + u k 1 ( x) 2 η 3 + u k+1 u k 1 + u k, 0 < ±η < x 2 x For this case, no general analytical formula can be obtained, as Θ(x k + ν, 0; x k )G(x k, x k + η) = e cη3 6ν e 1 4ν t [(1+b t)η2 +2a tν] For small t, we can use the approximation e cη3 6ν 1 cη 3 /6ν, so that the integration for u(x, t) can already be computed analytically.

12 Numerical schemes So, we have: with u(x k, t) = u(x k, 0) + s 1 t + s 2 t 2 + s 3 t 3 + s 4 t 4 k 0 + k 1 t + k 2 t 2 + k 3 t 3 + k 4 t 4 k 0 = 6ν, k 1 = 24bν, k 2 = (36b 2 + 6ac)ν, k 3 = a 3 c + (24b abc)ν, k 4 = a 3 bc + 6(b 4 + ab 2 c)ν, s1 = 6abν + 12cν 2, s 2 = ( 18ab 2 + 6a 2 c)ν + 24bcν 2, s3 = 18ab 3 ν + 12b 2 cν 2, a 4 bc 6(ab 4 + a 2 b 2 c)ν

13 Numerical schemes Scheme C: An extension of the scheme B, using piecewise quartic polynomial approximation by a five point formula, that results on: u(x k, t) = u(x k, 0)+ p 0 + p 1 t + p 2 t 2 + p 3 t 3 + p 4 t 4 + p 5 t 5 + p 6 t 6 q 0 + q 1 t + q 2 t 2 + q 3 t 3 + q 4 t 4 + q 5 t 5 + q 6 t 6

14 Outline 1 Cole-Hopf transformation 2 Numerical schemes 3 Some results

15 Some results We will make two numerical tests: { 1, x 0 P1: u(x, 0) = 0, x > 0 P2: u(x, 0) = 1 sin(πx)

16 Some results L 1 error at t = 1 versus x in P1 for ν = 0.1 and t = x 2. Scheme B and Scheme C are nearly second and four order approximations, whereas Scheme A error does not seem to converge.

17 Some results Scheme C ν = 10 3, t = 10, x = 0.1, x = 0.01 Scheme C ν = 10 3, t = 10, x = 10 3, t = 10 4

18 Some results Modification for Scheme B and Scheme C. We increase ν to C 1 x locally in the computation of u(x k, t) if the corresponding slopes b ± = ± u k±1 u k x satisfy b + b < 0 or b ± > C 2 x, for C 1, C 2 R +. ν = 10 5, t = 10, x = 0.1, t = 0.01, C 1 = 1, C 2 = 1.5

19 Some results ν = 10 3, t = 1, x = 0.02, t = 0.002, C 1 = 1, C 2 = 1.5

20 Bibliography K. Sakai and I. Kimura, A numerical scheme based on a solution of nonlinear advection-diffusion equations, Journal of Computational and Applied Mathematics, vol. 173, 2005, pp

21 Thanks for your attention!