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Numerical approximations of evolution problems with nonlocal diffusion

Silvia Sastre Gómez

Bosco García Archilla Jorge Galán Vioque

24 May, 2010

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Introduction

Nonlocal evolution problems ut(x, t) = (J ∗ u − u)(x, t) =

• RN J(x − y)u(y, t)dy − u(x, t)

where J : RN → RN is non-negative, radial, continuous function

• RN J(z)dz = 1.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Meaning of each term in the equation

ut(x, t) =

• RN J(x − y)u(y, t)dy − u(x, t)

u(x, t) is the density at the point x at time t.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Meaning of each term in the equation

ut(x, t) =

• RN J(x − y)u(y, t)dy − u(x, t)

u(x, t) is the density at the point x at time t. J(x − y) is the probability distribution of jumping from y to x.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Meaning of each term in the equation

ut(x, t) =

• RN J(x − y)u(y, t)dy − u(x, t)

u(x, t) is the density at the point x at time t. J(x − y) is the probability distribution of jumping from y to x.

• RN J(x − y)(u(y, t) − u(x, t))dy takes into account the

individuals arriving or leaving position x to other places.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

The Dirichlet problem

Consider de nonlocal problem: ut(x, t) =

• RN J(x − y)u(y, t)dy − u(x, t),

x ∈ Ω, t > 0, u(x, t) = 0, x / ∈ Ω, t > 0, u(x, 0) = u0(x), x ∈ Ω. In this problem diffusion takes place in the whole RN, with u = 0

• utside Ω.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

The Dirichlet problem

Theorem (asymptotic behaviour) If u0 is continuous, positive and bounded then there exists a positive constant C ∗ such that lim

t→∞ max x

|u(x, t) − C ∗eλ1tφ1(x)| = 0. where λ1 is the first eigenvalue associated to the operator Au(x, t) =

• RN J(x − y)u(y, t)dy − u(x, t)

and φ1 the eigenfunction associated to λ1

[1] Emmanuel Chasseigne, Manuela Chaves, Julio Rossi, J. Math. Pures Appl. 86 (2006) 271-291. Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Our aim

Discretize the Dirichlet problem with different methods using MATLAB, and see which is the most efficient.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Our aim

Discretize the Dirichlet problem with different methods using MATLAB, and see which is the most efficient. Evaluate how the discretizations reproduce the asymptotic behaviour of the solutions of the equation.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Discretization of the problem

We restrict ourselves to the 1-dimensional case, so that Ω = [a, b], a < b ∈ R. We consider a partition of [a, b] given by the N equally-spaced points.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Discretization of the problem

To discretize the operator of our problem Au =

• R J(x − y)u(y, t)dy − u(x, t) we have used the trapezoidal

rule (Au)i = h  J(xi − x1)u1 + J(xi − xN)uN 2 +

N−1

• j=2

J (xi − xj) uj  −ui.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Discretization of the problem

Operator written with matrices: Au = A u =        h        J1/2 J2 J3 ... JN/2 J2/2 J1 J2 ... JN−1/2 J3/2 J2 J1 ... JN−2/2 . . . . . . . . . ... . . . JN/2 JN−1 JN−2 ... J1/2        − IN        u. where {J1 = J(0), J2 = J(h), J3 = J(2h), ..., JN = J(Nh)}. u = (u1, u2, u3, . . . , uN−1, uN)T and uj = uj(t) ≈ u(xj, t).

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

MATLAB commands

Convolution, with the MATLAB command conv. FFT techniques , with the MATLAB commands fft and ifft.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

MATLAB commands / Convolution

Convolution or polynomial multiplication If u = (u1, . . . , un)T and v = (v1, . . . , vm)T are vectors whose components are polynomial coefficients, its convolution is the vector whose components are the coefficients of the polynomial given by its product. This resulting vector is of length n + m − 1. Looking at it by coordinates, if w =conv(u,v), then wk =

• j

uj vk−j+1, k = 1, 2, . . . , n + m − 1. p(x) = u1 + u2x + . . . + unxn

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

MATLAB commands / Convolution

Convolution or polynomial multiplication If u = (u1, . . . , un)T and v = (v1, . . . , vm)T are vectors whose components are polynomial coefficients, its convolution is the vector whose components are the coefficients of the polynomial given by its product. This resulting vector is of length n + m − 1. Looking at it by coordinates, if w =conv(u,v), then wk =

• j

uj vk−j+1, k = 1, 2, . . . , n + m − 1. p(x) = u1 + u2x + . . . + unxn q(x) = v1 + v2x + . . . + vmxm

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

MATLAB commands / Convolution

Convolution or polynomial multiplication If u = (u1, . . . , un)T and v = (v1, . . . , vm)T are vectors whose components are polynomial coefficients, its convolution is the vector whose components are the coefficients of the polynomial given by its product. This resulting vector is of length n + m − 1. Looking at it by coordinates, if w =conv(u,v), then wk =

• j

uj vk−j+1, k = 1, 2, . . . , n + m − 1. p(x) = u1 + u2x + . . . + unxn q(x) = v1 + v2x + . . . + vmxm p(x)q(x) = w1 + w2x + . . . + wn+m−1xn+m−1

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

MATLAB commands / FFT

Discrete Fourier Transform (DFT) Linear transformation in CN that associates with each vector f the vector c as follows, FN c = f       1 1 1 · 1 1 ω ω2 · ωn−1 1 ω2 ω4 · ω2(n−1) · · · · · 1 ωN−1 ω2(N−1) · ω(N−1)2             c0 c1 c2 · cN−1       =       f0 f1 f2 · fN−1       where ω = e

2πi N is the N−th root of unity. Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

MATLAB commands / FFT

Discrete Fourier Transform (DFT) Linear transformation in CN that associates with each vector f the vector c as follows, FN c = f       1 1 1 · 1 1 ω ω2 · ωn−1 1 ω2 ω4 · ω2(n−1) · · · · · 1 ωN−1 ω2(N−1) · ω(N−1)2             c0 c1 c2 · cN−1       =       f0 f1 f2 · fN−1       where ω = e

2πi N is the N−th root of unity.

The Discrete Fourier Transform of a vector f is c = 1

N ¯

FN f .

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

MATLAB commands / FFT

Fast Fourier Transform (FFT) The DFT, to find c = 1

N ¯

FN f requires N2 multiplications. Whereas, the FFT (Cooley and Tukey algorithm) finds c = 1

N ¯

FN f with O(N log(N)) multiplications.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Methods implemented

1 Convolution with help of a matrix-vector product. 2 Computing the convolution by means of MATLAB’s command

conv.

3 Computing the convolution by means of Fast Fourier

Transform (FFT) techniques.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Comparison of the methods

Elapsed time (The measured duration of an event) per number of nodes N for the different methods.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Asymptotic behaviour of the discretization

u(x, t) → C ∗eλ1tφ1(x), as t → ∞

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Asymptotic behaviour of the discretization

u(x, t) → C ∗eλ1tφ1(x), as t → ∞ Particular case For the function J(x) = e− x2

σ2 , Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Asymptotic behaviour of the discretization

u(x, t) → C ∗eλ1tφ1(x), as t → ∞ Particular case For the function J(x) = e− x2

σ2 , in the interval Ω = [0, 1] Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Asymptotic behaviour of the discretization

u(x, t) → C ∗eλ1tφ1(x), as t → ∞ Particular case For the function J(x) = e− x2

σ2 , in the interval Ω = [0, 1]

and initial data u0 = e−1/(sin(π(x−1/2)))2.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Asymptotic behaviour of the discretization

u(x, t) → C ∗eλ1tφ1(x), as t → ∞ Particular case For the function J(x) = e− x2

σ2 , in the interval Ω = [0, 1]

and initial data u0 = e−1/(sin(π(x−1/2)))2. The approximation we have obtained for λ1 is λ1 ≈ −0, 3818.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Asymptotic behaviour of the discretization

u(x, t) → C ∗eλ1tφ1(x), as t → ∞ Particular case For the function J(x) = e− x2

σ2 , in the interval Ω = [0, 1]

and initial data u0 = e−1/(sin(π(x−1/2)))2. The approximation we have obtained for λ1 is λ1 ≈ −0, 3818. We have calculated also an approximation of C ∗ C ∗ = |(u0, φ1)| (φ1, φ1) ≈ 4, 0781.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Inner product

(x, y) = 1 2x1y1 + 1 2xNyN +

N−1

• i=2

xiyi

(from the trapezoidal rule)

The matrix A is selfadjoint w.r.t. (·, ·). The eigenvectors of A are orthogonal w.r.t. (·, ·).

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Simulations

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Simulations

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Simulations

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Simulations

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Simulations

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Conclusions

The best method is the one that uses the FFT techniques.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Conclusions

The best method is the one that uses the FFT techniques. The worst method is the one that uses the MATLAB command conv.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Conclusions

The best method is the one that uses the FFT techniques. The worst method is the one that uses the MATLAB command conv. The asymptotic behaviour of the discretization seems to be well approximated.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Future work

Numerical approximation for larger spatial dimension.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Future work

Numerical approximation for larger spatial dimension. Numerical approximation for evolution problems with nonlocal diffusion, with Neumann boundary conditions. ut(x, t) =

• Ω

J(x − y)u(y, t)dy − u(x, t), x ∈ Ω, t > 0 u(x, 0) = u0(x), x ∈ Ω.

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion

Bibliography

[1] Emmanuel Chasseigne, Manuela Chaves, Julio Rossi, Asymptotic behaviour for nonlocal diffusion equations, J. Math. Pures Appl. 86 (2006) 271-291. [2] H. Brezis, A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math Pures Appl. 62 (1983) 73-97. [3] G. Strang, Introduction to Applied Mathematics, Wellesley-Cambridge

• Press. (1986).

[4] J.M. Sanz-Serna, Fourier Techniques in Numerical Methods for Evolutionary Problem, Dpto. de Matemática Aplicada y Computación, Facultad de Ciencias, Universidad de Valladolid, Spain. [5] Cooley, J. W. and Tukey, J. W., Math. Comput. 19 297 [6] Lloyd N. Trefethen, Spectral Methods in MATLAB, siam

Silvia Sastre Gómez Bosco García Archilla Jorge Galán Vioque Numerical approximations of evolution problems with nonlocal diffusion