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 � u t ( x , t ) = ( J ∗ u − u )( x , t ) = R N J ( x − y ) u ( y , t ) dy − u ( x , t ) where J : R N → R N is non-negative, radial, continuous function � R N 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 � u t ( x , t ) = R N 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 � u t ( x , t ) = R N 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 � u t ( x , t ) = R N 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 . � R N 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: � u t ( x , t ) = R N 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 ) = u 0 ( x ) , x ∈ Ω . In this problem diffusion takes place in the whole R N , with u = 0 outside 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 u 0 is continuous, positive and bounded then there exists a positive constant C ∗ such that | u ( x , t ) − C ∗ e λ 1 t φ 1 ( x ) | = 0 . t →∞ max lim x where λ 1 is the first eigenvalue associated to the operator � Au ( x , t ) = R N J ( x − y ) u ( y , t ) dy − u ( x , t ) and φ 1 the eigenfunction associated to λ 1 [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 N − 1 J ( x i − x 1 ) u 1 + J ( x i − x N ) u N � ( Au ) i = h + J ( x i − x j ) u j − u i . 2 j = 2 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: J 1 / 2 J 2 J 3 ... J N / 2 J 2 / 2 J 1 J 2 ... J N − 1 / 2 J 3 / 2 J 2 J 1 ... J N − 2 / 2 Au = A u = h − I N u . . . . . ... . . . . . . . . J N / 2 J N − 1 J N − 2 ... J 1 / 2 where { J 1 = J ( 0 ) , J 2 = J ( h ) , J 3 = J ( 2 h ) , ..., J N = J ( Nh ) } . u = ( u 1 , u 2 , u 3 , . . . , u N − 1 , u N ) T and u j = u j ( t ) ≈ u ( x j , 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 = ( u 1 , . . . , u n ) T and v = ( v 1 , . . . , v m ) 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 � w k = u j v k − j + 1 , k = 1 , 2 , . . . , n + m − 1 . j p ( x ) = u 1 + u 2 x + . . . + u n x n 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 = ( u 1 , . . . , u n ) T and v = ( v 1 , . . . , v m ) 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 � w k = u j v k − j + 1 , k = 1 , 2 , . . . , n + m − 1 . j p ( x ) = u 1 + u 2 x + . . . + u n x n q ( x ) = v 1 + v 2 x + . . . + v m x m 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 = ( u 1 , . . . , u n ) T and v = ( v 1 , . . . , v m ) 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 � w k = u j v k − j + 1 , k = 1 , 2 , . . . , n + m − 1 . j p ( x ) = u 1 + u 2 x + . . . + u n x n q ( x ) = v 1 + v 2 x + . . . + v m x m p ( x ) q ( x ) = w 1 + w 2 x + . . . + w n + m − 1 x n + 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 C N that associates with each vector f the vector c as follows, F N c = f 1 1 1 · 1 c 0 f 0 ω 2 ω n − 1 1 ω · c 1 f 1 ω 2 ω 4 ω 2 ( n − 1 ) 1 · c 2 = f 2 · · · · · · · ω ( N − 1 ) 2 ω N − 1 ω 2 ( N − 1 ) c N − 1 f N − 1 1 · 2 π i N is the N − th root of unity. where ω = e 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 C N that associates with each vector f the vector c as follows, F N c = f 1 1 1 · 1 c 0 f 0 ω 2 ω n − 1 1 ω · c 1 f 1 ω 2 ω 4 ω 2 ( n − 1 ) 1 · c 2 = f 2 · · · · · · · ω ( N − 1 ) 2 ω N − 1 ω 2 ( N − 1 ) c N − 1 f N − 1 1 · 2 π i N is the N − th root of unity. where ω = e N ¯ The Discrete Fourier Transform of a vector f is c = 1 F N 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) F N f requires N 2 multiplications. N ¯ The DFT, to find c = 1 N ¯ Whereas, the FFT (Cooley and Tukey algorithm) finds c = 1 F N 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
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