Meshfree Exponential Integrators
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1 Meshfree joint work with A. Ostermann (Innsbruck) M. Caliari (Verona) Leopold Franzens Universität Innsbruck Innovative Integrators 3 October 2 Meshfree
2 Problem class: Goal: Time-dependent PDE s with dominating advection part Solution has a small essential support High dimensional problems Develop an integrator with high accuracy both in space and time Use the special form of the solution to save memory Meshfree
3 Example: Consider the Molenkamp Crowley equation t u = x (au) + y (bu) with The initial profile, a(x, y) = 2πy, b(x, y) = 2πx. u (x, y) = exp( (x.2) 2 (y.2) 2 ), rotates around the origin. Meshfree
4 Meshfree
5 Given a stiff initial-value problem u (t) = F (u(t)), u() = u. Linearising the problem at a state w gives v (t) = Av(t) + g (v(t)), v() = u w with A = DF (w), v(t) = u(t) w and g a nonlinear reminder. Using the variation-of-constant formula the solution has the form v(t) = e t A v + t e (t τ)a g (v(τ))dτ Meshfree
6 Given a stiff initial-value problem u (t) = F (u(t)), u() = u. Linearising the problem at a state w gives v (t) = Av(t) + g (v(t)), v() = u w with A = DF (w), v(t) = u(t) w and g a nonlinear reminder. Using the variation-of-constant formula the solution has the form v(t) = e t A v + t e (t τ)a g (v(τ))dτ Meshfree
7 Example: Exponential Euler Method g (v(τ)) g (v ) gives v = e h A v + hϕ (h A)g (v ). Here h denotes the step size and ϕ is the entire function ϕ (z) = ez z. Exponential and related functions will be approximated with the Leja point method (based on Newton s interpolation formula at a Leja sequence). Meshfree
8 The Leja point method needs a rough approximation of largest eigenvalues of the Operator. This can be done by splitting the operator in a symmetric and skew-symmetric part and computing the largest in magnitude eigenvalues of both parts. We get estimate of the spectrum in form of a rectangle with vertices ( a, ib),(, ib),(,ib),( a,ib), a,b. Distinguish two cases: a b, take Leja points on interval [ a,]. a < b, consider conjugate pairs of Leja points on domain {z C: R(z) = a/2,i(z) [ b,b]} Meshfree
9 Some properties: similar distribution as Chebyshev points defined recursively - combines well with Newton interpolation based on matrix-vector multiplications superlinear convergence real arithmetic Meshfree
10 Basis function: Radial Basis Functions Meshfree
11 Given a function f : R d R sampled at point set X = {x,..., x m } Approximate f by the interpolant s(ξ) = λ x φ( ξ x ) x X using a radial function φ : R + R. The coefficients λ = (λ x ) x X are chose such that with A = {φ ( x i x j ) } xi,x j X. s X = f X = Aλ = f X Meshfree
12 RBF interpolation: works in any dimension simple to implement high accuracy Meshfree
13 Compactly supported RBFs Meshfree
14 We use Wendland functions. Defined via φ d,k = I k φ d/2 +k+, φ l (r ) = ( r ) l + where (I φ)(r ) = r tφ(t)dt and d is the space dimension. d φ d,k (r ) smoothness φ, (r ) = ( r ) + C φ, (r ) = ( r ) 3 + (3r + ) 2 C 3 φ 3, (r ) = ( r ) 2 + C φ 3, (r ) = ( r ) 4 + (4r + ) 2 C φ 3,2 (r ) = ( r ) 6 + (35r 2 + 8r + 3) C 4 φ 3,3 (r ) = ( r ) 8 + (32r r 2 + 8r + ) C 6 Meshfree
15 Some properties of φ d,k (r ) smoothness 2k strictly positive definite on R d local interpolation error bounded by f (y) s(y) c f Ch k+/2 ρ, h ρ < h for h small enough, where h ρ (y) := max y Bρ (ξ) min x X ξ x 2 for a given ρ >. condition number of interpolation matrix A bounded by cond 2 (A) C q d 2k x, { } where q X = /2min xi,x j X xi x j 2 x i x j. Meshfree
16 Consider a linear differential equation discretized at X u t (t, ) X = Lu(t, ) X, u(, ) X = u X. Approximate Lu(t,ξ) by Ls(t,ξ) = L λ x (t)φ( ξ x ) = λ x (t)lφ( ξ x ). x X x X This gives s t (t, ) X = A L λ with A L = {Lφ ( x i x j ) } xi,x j X and λ = A u(, t) X. Meshfree
17 Consider a linear differential equation discretized at X s t (t, ) X = Ls(t, ) X, s(,) X = u X. Approximate Lu(t,ξ) by Ls(t,ξ) = L λ x (t)φ( ξ x ) = λ x (t)lφ( ξ x ). x X x X This gives s t (t, ) X = A L A s(t, ) X, with A L = {Lφ ( x i x j ) } xi,x j X. Meshfree
18 Standard forward difference approximation in time leads to a simple integrator s(t n+, ) X = s(t n, ) X + t A L A s(t n, ) X. Meshfree
19 For the error estimate in space use two set of points: interpolation points check points Every interpolation point has some checkpoints. Suppose f is known everywhere. Idea: interpolate at interpolation points evaluate at check points calculate error at check points to get error estimate in space Refinement If error at a check point is too large, check point becomes interpolation point. Coarsening If error at all check points corresponding to a interpolation point are small enough, interpolation point will be removed. Meshfree
20 Meshfree
21 Meshfree
22 Meshfree
23 Meshfree
24 Meshfree
25 Meshfree
26 Meshfree
27 Meshfree
28 Meshfree
29 Meshfree
30 Meshfree
31 check points For a given set of interpolation points calculate a Delaunay triangulation Take circumcenters of resulting triangles as check points. Two reasons Check points maximize the local error bound If a refinement is needed, adding such a check point minimizes the growth or condition number of interpolation matrix. Meshfree
32 Example: Consider the Molenkamp Crowley equation t u = x (au) + y (bu) with a(x, y) = 2πy, b(x, y) = 2πx and u (x, y) = exp( (x.2) 2 (y.2) 2 ). Meshfree
33 The initial profile rotates around the origin. Tolerances: Space: 3 ; Time: 3 ; constant time step size: 5 2 Compute 4 turns of the pulse (8 steps) Meshfree
34 2 exact numeric Meshfree
35 Two numerical experiments: achieved accuracy (for prescribed tolerances) long time computation Step size for time evolution: 2. Meshfree
36 Error at T = error tolerance Meshfree
37 Required number of basis functions 4 number of points tolerance Meshfree
38 Long time computation ( turns), tolerance = 5 2 t= x 6 2 t= 5 x 3 2 t=3 2 x t=5 x 3 2 t=8 x 3 2 t= Meshfree
39 Global error 2 error turns Meshfree
40 Number of basis functions number of points time Meshfree
41 Example2: Consider the semilinear advection-diffusion-reaction problem ( ) t u = /2 xx u + y y u + 6 ( x u + y u ) + 7u (u /2)( u) Integrated with exponential Rosenbrock 3(2) up to T =.. Meshfree
42 Meshfree
43 Global error at T =. error anticipated err. global err. tolerance tolerance Meshfree
44 Number of time steps and number of interpolation points 4 number of points number of steps tolerance Meshfree
45 Development of interpolation point set Integrate advection-diffusion-reaction problem up to T =.3, with tolerance 5 4 Meshfree
46 Meshfree
47 Development of interpolation point set 2 t= 2 t=.64 2 t= t=.34 2 t=.97 2 t= Meshfree
48 4 35 number of points time Meshfree
49 Outlook efficient implementation error analysis solve real life problems Meshfree
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