Meshfree joint work with A. Ostermann (Innsbruck) M. Caliari (Verona) Leopold Franzens Universität Innsbruck Innovative Integrators 3 October 2 Meshfree
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
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
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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
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
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
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
Some properties: similar distribution as Chebyshev points defined recursively - combines well with Newton interpolation based on matrix-vector multiplications superlinear convergence real arithmetic Meshfree
Basis function: Radial Basis Functions Meshfree
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
RBF interpolation: works in any dimension simple to implement high accuracy.9.8.7.6.5.4.3.2..8.6.4.2.2.4.6.8 Meshfree
Compactly supported RBFs Meshfree
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 3 + 25r 2 + 8r + ) C 6 Meshfree
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
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
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
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
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
.8.6.4.2 2 2 2 2 Meshfree
.8.6.4.2 2 2 2 2 Meshfree
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.8.6.4.2 2 2 2 2 Meshfree
.8.6.4.2 2 2 2 2 Meshfree
.8.6.4.2 2 2 2 2 Meshfree
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.8.6.4.2 2 2 2 2 Meshfree
.6.4.2 2 2 2 2 Meshfree
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
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
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
2 exact numeric.3.2. 2 2 2. Meshfree
Two numerical experiments: achieved accuracy (for prescribed tolerances) long time computation Step size for time evolution: 2. Meshfree
Error at T = error 2 3 4 5 6 7 7 6 5 4 3 2 tolerance Meshfree
Required number of basis functions 4 number of points 3 2 7 6 5 4 3 2 tolerance Meshfree
Long time computation ( turns), tolerance = 5 2 t= x 6 2 t= 5 x 3 2 t=3 2 x 3 5 3 3 2 2 2 2 2 2 2 2 2 2 t=5 x 3 2 t=8 x 3 2 t=.2 6.8 4 6 2 2.4 2 2 2 2 2 2 2 2 2 Meshfree
Global error 2 error 3 4 2 turns Meshfree
Number of basis functions 45 44 number of points 43 42 4 4 39 38 2 4 6 8 time Meshfree
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
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Global error at T =. error 3 5 7 anticipated err. global err. tolerance 7 6 5 4 3 2 tolerance Meshfree
Number of time steps and number of interpolation points 4 number of points number of steps 3 2 6 5 4 3 2 tolerance Meshfree
Development of interpolation point set Integrate advection-diffusion-reaction problem up to T =.3, with tolerance 5 4 Meshfree
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Development of interpolation point set 2 t= 2 t=.64 2 t=.5 2 2 2 2 2 2 2 2 2 2 t=.34 2 t=.97 2 t=.25 2 2 2 2 2 2 2 2 2 Meshfree
4 35 number of points 3 25 2 5.5..5.2.25 time Meshfree
Outlook efficient implementation error analysis solve real life problems Meshfree