HOPS Tutorial: Lecture 1 Boundary Value Problems

Transcription

1 HOPS Tutorial: Lecture 1 Boundary Value Problems David P. Nicholls Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago HOPS Tutorial: Lecture 1 David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 1 / 45

2 Collaborators Collaborators: Fernando Reitich (Minnesota) Ben Akers (AFIT) Alison Malcolm (MIT) Jie Shen (Purdue) Nilima Nigam (Simon Fraser) Walter Craig (McMaster; Fields) Students: Mark Taber, Carlo Fazioli, Zheng Fang, Venu Tammali Thanks to: NSF (DMS ) David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 2 / 45

3 References Milder, JASA 89(2): (1991). Milder, Radio Sci. 31(6): (1996). Bruno & Reitich, JOSA 10(6): (1993). Bruno & Reitich, JOSA 10(11): (1993). Bruno & Reitich, JOSA 10(12): (1993). Craig & Sulem, JCP 108(1): (1993). DPN & Reitich, PRSE A 131(6): (2001). DPN & Reitich, JCP 170(1): (2001). DPN & Reitich, Numer. Math. 94(1): (2003). David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 3 / 45

4 Introduction Numerical Solution of PDE Partial Differential Equations (PDEs) have long been recognized as the most powerful and successful mathematical modeling tool for engineering and science, and the study of surface water waves is no exception. With the advent of the modern computer, the possibility of numerical simulation of PDEs at last became a practical reality. The last years has seen an explosion in the development and implementation of algorithms which are: Robust, highly accurate, and fast. Among the myriad choices are 1 Finite Difference Methods 2 Finite Element Methods (Continuous and Discontinuous) 3 High Order Spectral (Element) Methods 4 Boundary Integral/Element Methods 5 Many more... David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 4 / 45

5 Introduction Numerical Simulation of Interfacial PDEs The class of High Order Perturbation of Surfaces (HOPS) methods we describe here are particularly well suited for PDEs posed on piecewise homogeneous domains. Such layered media problems abound in the sciences: Free surface fluid mechanics (e.g., the water wave problem), Acoustic waves in piecewise constant density media, Electromagnetic waves interacting with grating structures, Elastic waves in sediment layers. For such problems these HOPS methods can be: highly accurate (error decaying exponentially as the number of degrees of freedom increases), rapid (an order of magnitude fewer unknowns as compared with volumetric formulations), robust (delivering accurate results for rather rough/large interface shapes). These HOPS schemes are not competitive for problems with inhomogeneous domains and/or extreme geometries. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 5 / 45

6 Laplace s Equation Water Wave Problem: Gravity Waves on Deep Water To fix on a problem we consider a water wave problem: Model the evolution of the free surface of a deep, two dimensional, ideal fluid under the influence of gravity. The widely accepted model (Lamb; 1879,...,1932) is In these ϕ = 0, y < η(x, t) y ϕ 0, y t η + x η( x ϕ) = y ϕ, y = η(x, t) t ϕ + (1/2) ϕ ϕ + gη = 0, ϕ(x, y, t) is the velocity potential, u = ϕ, η(x, t) is the air water interface, g is the gravitational constant. y = η(x, t). David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 6 / 45

7 Laplace s Equation The Dirichlet Neumann Operator (DNO) At the center of this problem is the solution of the Elliptic Boundary Value Problem (BVP) v = 0 y v 0 v = ξ y < g(x) y y = g(x). In particular, with the Dirichlet Neumann Operator (DNO) G(g)[ξ] = [ y v ( x g) x v] y=g(x), the water wave problem becomes (Zakharov 1968; Craig & Sulem 1993) t η = G(η)ξ, t ξ = gη A(η)B(η, ξ) [ ( A = ( x η) 2)] 1, B = ( x ξ) 2 (G(η)ξ) 2 2( x η)( x ξ)(g(η)ξ). David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 7 / 45

8 Laplace s Equation Lateral Boundary Conditions: Periodicity For many problems of practical interest it suffices to consider the classical periodic boundary conditions, e.g., v(x + L, y) = v(x, y), g(x + L) = g(x), L = 2π. In this case we can express functions in terms of their Fourier Series g(x) = ĝ p e ipx, ĝ p = 1 2π g(s)e ips ds. 2π 0 p= Thus, we focus on the BVP: v = 0 y v 0 v = ξ v(x + 2π, y) = v(x, y). y < g(x) y y = g(x) David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 8 / 45

9 Field Expansions The Method of Field Expansions (FE) Our first HOPS approach for DNOs solves the BVP directly. Its orgins: The work of Rayleigh (1907) and Rice (1951). The first high order implementation is due to Bruno & Reitich (1993; 3 papers) and was originally denoted the Method of Variation of Boundaries. To prevent confusion with subsequent methods it was later renamed the Method of Field Expansions (FE). The key to the method: The realization that interior to the domain (i.e., y < g ) the solution of Laplace s equation is v(x, y) = a p e p y e ipx. p= This HOPS approach uses the fact that, for a boundary perturbation g(x) = εf (x), the field, v = v(x, y; ε), depends analytically upon ε. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 9 / 45

10 Field Expansions The FE Method Assume that the interface is shaped by g(x) = εf (x) where f O (1) and, initially, ε 1. We will be able to show a posteriori that v depends analytically upon ε so that v = v(x, y; ε) = v n (x, y)ε n. n=0 Inserting this expansion into the governing equations and equating at orders O (ε n ) yields v n = 0 y < 0 y v n 0 y v n = Q n y = 0 v n (x + 2π, y) = v n (x, y). David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 10 / 45

11 Field Expansions The FE Method: Solutions The crucial term is the boundary inhomogeneity Q n (x) = δ n,0 ξ(x) n 1 m=0 This form comes from the expansion v(x, εf ; ε) = v n (x, εf )ε n = n=0 F n m (x) n m y v m (x, 0), F m (x) := f m (x) m! n=0 ε n n m=0 F n m (x) n m y v m (x, 0). The solution of Laplace s equation subject to the bottom and lateral boundary conditions is v n (x, y) = p= a n,p e p y e ipx. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 11 / 45

12 FE Recursions Field Expansions Inserting this form into the surface boundary condition delivers p= a n,p e ipx = p= where, since exp( p εf ) = m εm F m p m, Q n (x) = δ n,0 p= ˆξ p e ipx n 1 m=0 F n m (x) More precisely we have the FE Recursions a n,p = δ n,0 ˆξ p n 1 m=0 q= ˆQ n,p e ipx, p= p n m a m,p e ipx, ˆF n m,p q q n m a m,q. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 12 / 45

13 The DNO via FE Field Expansions The FE recursions deliver the solution everywhere well inside the problem domain. However, is the expansion v(x, y) = a p e p y e ipx p= valid at the boundary? Near the boundary? Assuming that there is some validity at the boundary, recall that we wish to compute the Neumann data Expanding in ε ν n (x)ε n = n=0 ν(x) = [ y v ( x g) x v] y=g(x). [ y v n (x, εf ) ε( x f ) x v n (x, εf )] ε n. n=0 David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 13 / 45

14 Field Expansions FE Recursions for the DNO Equating at order O (ε n ) gives ν n (x) = n m=0 F n m n+1 m y Once again, more precisely, ˆν n,p = n m=0 q= n 1 n 1 v m (x, 0) ( x f )F n 1 m x y n 1 m v m (x, 0). m=0 ˆF n m,p q q n+1 m a m,q m=0 q= where F m(x) := ( x f )F m (x). ˆF n 1 m,p q (iq) q n 1 m a m,q, David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 14 / 45

15 Field Expansions MATLAB Code for FE Recursions: Field function [anp] = field_fe_laplace(xi,f,p,nx,n) anp = zeros(nx,n+1); fn = zeros(nx,n+1); fn(:,0+1) = ones(nx,1); for n=1:n fn(:,n+1) = f.*fn(:,n-1+1)/n; end anp(:,0+1) = fft( xi ); for n=1:n anp(:,n+1) = 0*xi; for m=0:n-1 anp(:,n+1) = anp(:,n+1) - fft(fn(:,n-m+1).*... ifft(abs(p).^(n-m).*anp(:,m+1))); end end David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 15 / 45

16 Field Expansions MATLAB Code for FE Recursions: DNO function [Gn] = dno_fe_laplace(anp,f,p,nx,n) Gn = zeros(nx,n+1); fn = zeros(nx,n+1); fn(:,0+1) = ones(nx,1); for n=1:n fn(:,n+1) = f.*fn(:,n-1+1)/n; end f_x = ifft( (1i*p).*fft(f) ); for n=0:n Gn(:,n+1) = ifft( abs(p).*anp(:,n+1) ); for m=0:n-1 Gn(:,n+1) = Gn(:,n+1) + fn(:,n-m+1).*... ifft( abs(p).^(n+1-m).*anp(:,m+1) )... - f_x.*fn(:,n-1-m+1).*ifft( (1i*p).*... abs(p).^(n-1-m).*anp(:,m+1) ); end end David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 16 / 45

17 Operator Expansions The Method of Operator Expansions (OE) The second HOPS approach we investigate considers the DNO alone without explicit reference to the underlying field equations. For this reason the method has been termed the Method of Operator Expansions (OE). The first high order implementation for electromagnetics (the Helmholtz equation) is due to Milder (1991; two papers) and Milder & Sharp (1992; two papers). The first high order implementation for water waves (the Laplace equation) is due to Craig & Sulem (1993). Once again, we use, in a fundamental way, the representation v(x, y) = a p e p y e ipx. p= This HOPS method uses the fact that, for a boundary perturbation g(x) = εf (x), the DNO, G = G(εf ), depends analytically upon ε. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 17 / 45

18 Operator Expansions The OE Method Again, assume that the interface is shaped by g(x) = εf (x) where f O (1) and, initially, ε 1. We now focus on the definition of the DNO, G, G(g)[ξ] = ν, and seek the action of G on a basis function, exp(ipx). To achieve this we use a solution of Laplace s equation (and relevant boundary conditions): v p (x, y) := e p y e ipx. Inserting the solution v p (x, y) into the defintion of the DNO gives G(g)[v p (x, g(x))] = [ y v p ( x g) x v p ] y=g(x). David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 18 / 45

19 Operator Expansions OE Recursions: Order Zero Assume that everything is analytic in ε and expand ( ) [ ] ε n G n (f ) ε m F m p m e ipx = n=0 m=0 ε n F n p n+1 e ipx n=0 ε( x f ) ε n F n (ip) p n e ipx. At O ( ε 0) this reads G 0 [ e ipx ] = p e ipx, so that we can conclude that G 0 [ξ] = G 0 [ p= ˆξ p e ipx ] = p= ˆξ p G 0 [ e ipx] = n=0 p= p ˆξ p e ipx =: D ξ. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 19 / 45

20 Operator Expansions OE Recursions: Higher Order Corrections At order O (ε n ), n > 0, we find n m=0 [ G m (f ) F n m p n m e ipx] = F n p n+1 e ipx ( x f )F n 1 (ip) p n 1 e ipx, which we can write as [ G n (f ) e ipx] = F n p n+1 e ipx ( x f )F n 1 (ip) p n 1 e ipx n 1 m=0 [ G m (f ) F n m p n m e ipx]. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 20 / 45

21 OE Recursions, cont. Operator Expansions The Fourier multiplier D m is defined by D m e ipx = p m e ipx, so that [ G n (f ) e ipx] = F n D n+1 e ipx ( x f )F n 1 x D n 1 e ipx n 1 m=0 [ G m (f ) F n m D n m e ipx]. Since (ip) 2 = p 2 we deduce that D 2 = 2 x and we arrive at [ G n (f ) e ipx] ) = ( F n x 2 ( x f )F n 1 x D n 1 e ipx n 1 m=0 [ G m (f ) F n m D n m e ipx]. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 21 / 45

22 Operator Expansions Slow OE Recursions Next, since x [F n x G] = F n 2 x G + ( x f )F n 1 x G, we have [ G n (f ) e ipx] = x F n x D n 1 e ipx n 1 m=0 [ G m (f ) F n m D n m e ipx]. As we have the action of G n we write down the Slow OE Recursions G n (f ) [ξ] = x F n x D n 1 ξ n 1 m=0 G m (f ) [ F n m D n m ξ ]. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 22 / 45

23 Operator Expansions Accelerating the OE Recursions: Adjointness So what is wrong with this set of recursions? To compute G n one must evaluate G n 1, which requires the application of G n 2, etc. Since the argument of G m changes as m changes, these cannot be precomputed and stored. Therefore, a naive implementation will require time proportional to O (n!). One can improve this by storing G m as an operator (a matrix in finite dimensional space). To compute G n requires time proportional to O ( nn 2 x ). Happily we can do even better by using the self adjointness properties of the DNO. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 23 / 45

24 Operator Expansions Self Adjointness of the DNO: Fast Recursions It can be shown that the DNO, G, and all of its Taylor series terms G n are self adjoint: G = G and G n = G n. Question: How can we use this to advantage? Answer: Recall that (AB) = B A, x = x, and F n = F n ; now take the adjoint of G n. Thus we realize the Fast OE Recursions G n (f ) [ξ] = D n 1 x F n x ξ n 1 m=0 D n m F n m G m (f ) [ξ]. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 24 / 45

25 Operator Expansions MATLAB Code for OE Recursions function [Gn] = dno_oe_laplace(xi,f,p,nx,n) Gn = zeros(nx,n+1); fn = zeros(nx,n+1); fn(:,0+1) = ones(nx,1); for n=1:n fn(:,n+1) = f.*fn(:,n-1+1)/n; end xi_hat = fft(xi); xi_x = ifft( (1i*p).*xi_hat ); Gn(:,0+1) = ifft( abs(p).*xi_hat ); for n=1:n Gn(:,n+1) = -ifft( abs(p).^(n-1).*(1i*p).*... fft(fn(:,n+1).*xi_x) ); for m=0:n-1 Gn(:,n+1) = Gn(:,n+1) - ifft( abs(p).^(n-m).*... fft(fn(:,n-m+1).*gn(:,m+1)) ); end end David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 25 / 45

26 Numerical Tests Numerical Tests Now that we have two HOPS schemes for approximating DNOs, we should test them... and play! To test, it would be ideal to have an exact solution... in fact we do! Recall the solution we used for the OE formula v p (x, y) := e p y e ipx. If we choose a wavenumber, say r, and a profile f (x), for a given ε > 0, it is easy to see that the Dirichlet data generates Neumann data ξ r (x; ε) := v r (x, εf (x)) = e r εf (x) e irx. ν r (x; ε) := [ y v r ε( x f ) x v r ] (x, εf (x)) = [ r ε( x f )(ir)] e r εf (x) e irx. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 26 / 45

27 Convergence of FE Numerical Tests 10 2 Relative Error versus N FE(Taylor) Relative Error Interface shape: f (x) = exp(cos(x)) Physical parameters: L = 2π, ε = Numerical parameters: N x = 64, N = N David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 27 / 45

28 Convergence of OE Numerical Tests 10 2 Relative Error versus N OE(Taylor) Relative Error Interface shape: f (x) = exp(cos(x)) Physical parameters: L = 2π, ε = Numerical parameters: N x = 64, N = N David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 28 / 45

29 Numerical Tests Comparison of FE and OE Relative Error Relative Error versus N FE(Taylor) OE(Taylor) Interface shape: f (x) = exp(cos(x)) Physical parameters: L = 2π, ε = Numerical parameters: N x = 64, N = N David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 29 / 45

30 Three Dimensions Three Dimensions A generalization of crucial importance is to the more realistic situation of a genuinely three dimensional fluid. In this case the air fluid interface, y = g(x) = g(x 1, x 2 ) is two dimensional rather than one dimensional. Such a generalization for Boundary Integral/Element Methods requires a new formulation as the fundamental solution changes, Φ 2 (r) = C 2 ln(r), Φ 3 (r) = C 3 r 1. One of the most appealing features of our HOPS methods is the trivial nature of the changes required moving from two to three dimensions. This can be seen by inspecting the solution of Laplace s equation: v(x, y) = a p e p y e ip x, p = (p 1, p 2 ) Z 2. p =0 David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 30 / 45

31 Three Dimensions Field Expansions in Three Dimensions Once again assuming g(x 1, x 2 ) = εf (x 1, x 2 ), f O (1), ε 1, we expand v = v(x, y; ε) = v n (x, y)ε n. n=0 We find the same inhomogeneous Laplace problem at every perturbation order n which has solution v n (x, y) = a n,p e p y e ip x. p =0 Following the development from before we find a n,p = δ n,0 ˆξp n 1 m=0 q =0 ˆF n m,p q q n m a m,q. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 31 / 45

32 Three Dimensions FE Recursions for the DNO: Three Dimensions As before, if we seek the Neumann data and expand ν(x) = [ y v ( x g) x v] y=g(x), ν(x; ε) = ν n (x)ε n, n=0 then FE delivers approximations to ν n from the a n,p. More precisely, for F m(x) := ( x f )F m (x), ˆν n,p = n m=0 q =0 n 1 ˆF n m,p q q n+1 m a m,q m=0 q =0 ˆF n 1 m,p q (iq) q n 1 m a m,q, David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 32 / 45

33 Three Dimensions The OE Method: Three Dimensions Again, assume that the interface is shaped by g(x 1, x 2 ) = εf (x 1, x 2 ), f O (1), ε 1. Once again we seek the action of G on a basis function, exp(ip x). To achieve this we use a solution of Laplace s equation (and relevant boundary conditions): We make the expansion and seek forms for the G n. v p (x, y) := e p y e ip x. G(εf ) = G n (f )ε, n=0 David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 33 / 45

34 Three Dimensions Operator Expansions: Three Dimensions Using the methods outlined ealier, we can show that and, for n > 0, G 0 [ξ] = D ξ = p ˆξ p e ip x, p =0 ] n 1 G n (f ) [ξ] = div x [F n x D n 1 ξ m=0 G m (f ) [ F n m D n m ξ ]. Again, these can be accelerated by adjointness considerations to G n (f ) [ξ] = D n 1 div x [F n x ξ] n 1 m=0 D n m F n m G m (f ) [ξ]. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 34 / 45

35 Finite Depth Finite Depth As we will now show, the generalization to finite depth is immediate. The problem statement is the same, save we replace y v 0 with y v(x, h) = 0. Once again, the key is the solution of Laplace s equation v p (x, y) := cosh( p (y + h)) e ipx. cosh( p h) Since v p and y v p evaluated at y = 0 will clearly become important, we introduce the symbol { 1 n even T n,p = T n,p (h) := tanh(h p ) n odd. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 35 / 45

36 FE: Finite Depth Finite Depth We can show that the FE Recursions become and that ˆν n,p = a p,n = δ n,0 ˆξ p n 1 n m=0 q= m=0 q= n 1 m=0 q= ˆF n m,p q q n m T n m,q (h)a m,q. ˆF n m,p q q n+1 m T n+1 m (h)a m,q ˆF n 1 m,p q (iq) q n 1 m T n 1 m (h)a m,q, David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 36 / 45

37 OE: Finite Depth Finite Depth For the OE recursions we can show that, at order zero, G 0 [ξ] = D tanh(h D )ξ, while at higher orders (after appealing to self adjointness) G n (f ) [ξ] = D n 1 T n 1,D x F n x ξ n 1 m=0 D n m T n m,d F n m G m (f ) [ξ]. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 37 / 45

38 Padé Summation Padé Summation A classical problem of numerical analysis: The approximation of an analytic function given a truncation of its Taylor series. Simply evaluating the truncation will be spectrally accurate in the number of terms for points inside the disk of analyticity. However, one may be interested in points of analyticity outside this disk. Thus, a numerical analytic continuation is of great interest. Padé approximation (Baker & Graves Morris, 1996) is one of the most popular choices. See also 8.3 of Bender & Orszag, David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 38 / 45

39 Taylor Summation Padé Summation Consider the analytic function c(ε) = c n ε n n=0 which we approximate by its truncated Taylor series c N (ε) := N c n ε n. n=0 If ε 0 is in the disk of convergence then c(ε 0 ) c N (ε 0 ) < K ρ N. However, if ε 0 is a point of analyticity outside the disk of convergence of the Taylor series, c N will produce meaningless results. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 39 / 45

40 Padé Summation Rational Approximation Padé approximation: Approximate c N by the rational function where L + M = N and [L/M](ε) := al (ε) b M (ε) = [L/M](ε) = c N (ε) + O L l=0 a lε l M m=0 b mε m ( ε L+M+1). We choose the equiorder Padé approximant [N/2, N/2](ε). For the purposes of the following discussion we assume that either N is even or, in the case N odd, that the highest order term c N is ignored so that L = M = N/2 is an integer. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 40 / 45

41 The Padé Algorithm Padé Summation To derive equations for the {a l } and {b m } we note that a M (ε) b M (ε) = c2m (ε) + O (ε 2M+1) is equivalent to b M (ε)c 2M (ε) = a M (ε) + O ( ε 2M+1), or ( M ) ( 2M ) ( M ) ( b m ε m c n ε n = a m ε m + O ε 2M+1). m=0 n=0 m=0 Multiplying the polynomials on the left hand side and equating at equal orders in ε m for 0 m 2M reveals two sets of equations m c m j b j = a m, j=0 [0 m M]; M c m j b j = 0, j=0 [M+1 m 2M]. Given that the c m are known, if the b m can be computed then the first equation can be used to find the a m. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 41 / 45

42 Padé Summation Padé Approximation, cont. Without loss of generality we make the classical specification b 0 = 1 so that the second equation becomes M c m j b j = c m, M + 1 m 2M, j=1 or H b = r where c M... c 1 b 1 c M+1... c 2 H =....., b 2 b =., r = c 2M 1... c M b M c M+1 c M+2. c 2M. David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 42 / 45

43 Padé Summation Taylor: Divergence of FE 10 1 Relative Error versus N FE(Taylor) Relative Error Interface shape: f (x) = exp(cos(x)) Physical parameters: L = 2π, ε = 0.2. Numerical parameters: N x = 256, N = N David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 43 / 45

44 Padé Summation Taylor: Divergence of OE 10 4 Relative Error versus N OE(Taylor) Relative Error Interface shape: f (x) = exp(cos(x)) Physical parameters: L = 2π, ε = 0.2. Numerical parameters: N x = 256, N = N David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 44 / 45

45 Padé Summation Padé: Convergence of FE and OE Relative Error Relative Error versus N FE(Taylor) FE(Pade) OE(Taylor) OE(Pade) Interface shape: f (x) = exp(cos(x)) Physical parameters: L = 2π, ε = 0.2. Numerical parameters: N x = 256, N = N David P. Nicholls (UIC) HOPS Tutorial: Lecture 1 HOPS Tutorial: Lecture 1 45 / 45