Chebfun2: Exploring Constant Coefficient PDEs on Rectangles

Transcription

1 Chebfun2: Exploring Constant Coefficient PDEs on Rectangles Alex Townsend Oxford University with Sheehan Olver, Sydney University FUN13 workshop, 11th of April 2013 Alex Townsend Constant coefficient PDEs 0/11

2 A simplified history of the Chebfun project Piecewise smooth (2010) Endpoint singularities (2010*) Reducing regularity Blow up functions (2011) Chebfun (2004) Linear ODEs (2008) Ṅonlinear ODEs (2012) Chebgui (2011*) Ordinary differential equations Chebfun2 (2013) Piecewise smooth Linear PDEs Arbitrary domains Nonlinear PDEs Two dimensions Alex Townsend Constant coefficient PDEs 1/11

3 Chebop and Chebop2: Overall workflow L = chebop(@(x,u) diff(u,2)-x*u, [-30 30]); Llbc = 1; Lrbc = 0; u = L \ 0; % Airy eqn % bcs % u = chebfun Convert handle into discretisation instructions Construct discretisation A R n n Impose bcs and solve Ãx = b Converged to the solution? yes Construct a chebfun no, increase n [Driscoll, Bornemann, & Trefethen, 2008], [Birkisson & Driscoll, 2012] Alex Townsend Constant coefficient PDEs 2/11

4 Chebop and Chebop2: Overall workflow L = chebop(@(x,u) diff(u,2)-x*u, [-30 30]); Llbc = 1; Lrbc = 0; u = L \ 0; % Airy eqn % bcs % u = chebfun Convert handle into discretisation instructions Construct discretisation A R n n Impose bcs and solve Ãx = b Converged to the solution? yes Construct a chebfun no, increase n [Driscoll, Bornemann, & Trefethen, 2008], [Birkisson & Driscoll, 2012] L = chebop2(@(u) diffx(u,2)+diffy(u,2)+100*u); Llbc = 1; Lrbc = 1; Lubc = 1; Ldbc = 1; u = L \ 0; % Helmholtz % bcs % u = chebfun2 Convert handle into discretisation instructions Construct a m by n generalised Sylvester matrix equation Impose bcs and solve matrix equation Converged to the solution? yes Construct a chebfun2 no, increase m or n or both Alex Townsend Constant coefficient PDEs 2/11

5 The numerical rank of a matrix, function, and operator For an m n matrix, Numerical Rank A = min(m,n) j=1 σ j u j v T j k σ j u j vj T, σ k+1 < tol For a function of two variables (this is what Chebfun2 uses), j=1 f (x, y)! = k σ j u j (y)v j (x) σ j u j (y)v j (x) j=1 j=1 For operators acting on functions of two variables, k L =! σ j L y j L x j σ j L y j L x j j=1 j=1 Discretise by a 1D spectral method [Olver & T, 2013] Alex Townsend Constant coefficient PDEs 3/11

6 Determining the rank of a PDE operator Given a PDE with constant coefficients of the form, L = N N j j=0 i=0 a ij i+j i x j y, A = (a ij), We can rewrite this as Find by AD, thanks A Birkisson 0 / x 0 L = D(y) T AD(x), D(x) = N / x N Hence, if A = UΣV T is the truncated SVD of A then ( ) ( ) ( ) L = D(y) T UΣV T D(x) = D(y) T U Σ V T D(x) This low rank representation for L tells us how to discretize and solve the PDE Alex Townsend Constant coefficient PDEs 4/11

7 Examples Helmholtz is of rank 2, u xx + u yy + 100u = (u xx + 50u) + (u yy + 50u) The SVD obtains the rank 2 expression, u xx + u yy + 100u = ( 0009u yy 0999u ) ( 0009u xx 0999u) ( 0999u yy 0009u ) (0009u xx 0999u) Rank 1: ODEs Trivial PDEs Rank 3: Almost everything else Rank 2: Laplace Helmholtz Transport Heat Wave The method of separation of variables is useful Alex Townsend Constant coefficient PDEs 5/11

8 Construct a m by n generalised Sylvester matrix equation If the PDE is Lu = f, where L is of rank-k then we solve for X R m n in, k σ j A j XBj T = F, A j R m m, B j R n n j=1 solution s ċoefficients 1D spectral ḋiscretization of Ly j, Lx j A j = ( ) Alex Townsend Constant coefficient PDEs 6/11

9 Matrix equation solvers Rank-1: A 1 XB1 T = F Solve A 1Y = F, then B 1 X T = Y T Rank-2: A 1 XB1 T + A 2XB2 T = F Generalised Bartels Stewart algorithm [Gardener, Laub, Amato, & Moler, 1992] Rank-k, k 3: Solve mn mn system with \ in Matlab 10 2 Time to solve matrix equation O(m 2 n 2 ) O(m 3 + n 3 ) Solve time O(mn) k = 1 k = 2 k m & n Alex Townsend Constant coefficient PDEs 7/11

10 Have we converged? (simplified version) Increase m and n until the tail of the solution s Chebyshev coefficients is near/below machine precision, and then truncate Always 2 j + 1 Truncated coefficients These coefficients can then be used by Chebfun2 to construct a low rank function approximation [T & Trefethen, 2013] Alex Townsend Constant coefficient PDEs 8/11

11 Overview and recap L = chebop2(@(u) diffx(u,2)+diffy(u,2)+100*u); Llbc = 1; Lrbc = 1; Lubc = 1; Ldbc = 1; u = L \ 0; % Helmholtz % bcs % u = chebfun2 Convert handle into discretisation instructions Construct a m by n generalised Sylvester matrix equation Impose bcs and solve matrix equation Converged to the solution? yes Construct a chebfun2 no, increase m or n or both Alex Townsend Constant coefficient PDEs 9/11

12 Demo DEMO Alex Townsend Constant coefficient PDEs 10/11

13 Future Work I would like to be able to do vector-valued PDEs such as Stokes equation: For Ω = [ 1, 1] [ 1, 1], u + p = f in Ω, u = 0 in Ω, u = 0 on Ω, p(x, y)dxdy = 0 in Ω Aiming for the following Chebfun2 syntax: N = chebop2v(@(u,p) [-lap(u) + grad(p); div(u)]); Nlbc = 0; Nrbc = 0; Nubc = 0; Ndbc = 0; Nbc sum2(p); uu = N \ [f;0]; Alex Townsend Constant coefficient PDEs 11/11

14 References [Birkisson & Driscoll, 2012] Automatic Fréchet differentiation for the numerical solution of boundary-value problems, (ACM Trans Math Softw, 2012) [Driscoll, Bornemann, & Trefethen, 2008] The chebop system for automatic solution of differential equations (BIT Numer Math, 2008) [Gardener, Laub, Amato, & Moler, 1992] Solution of the Sylvester matrix equation AXB T + CXD T = E, (ACM Trans Math Softw, 1992) [Olver & T, 2013] A fast and well-conditioned spectral method, (to appear in SIAM Review, 2013) [T & Trefethen, 2013] An extension of Chebfun to two dimensions, (submitted, 2013) Alex Townsend Constant coefficient PDEs 12/11