Atmospheric Dynamics with Polyharmonic Spline RBFs

Transcription

1 Photos placed in horizontal position with even amount of white space between photos and header Atmospheric Dynamics with Polyharmonic Spline RBFs Greg Barnett Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy s National Nuclear Security Administration under contract DE-NA SAND C

2 Outline Polyharmonic Spline (PHS) RBFs with Polynomials 1D Example Interpolation and Differentiation Weights Interpolation/Weights in D Weights on the Sphere Semi-Lagrangian Transport Governing Equations Limiter/Fixer Test Cases Results Eulerian Shallow Water Model Governing Equations Test Cases Results Conclusions and Future Work

3 Table of RBFs φ x, x R d, ε R Name (acronym) 1 + ε x Multiquadric (MQ) 1 Inverse Quadratic (IQ) 1 + ε x ε x Inverse Multiquadric (IMQ) e ε x Gaussian (GA) x k+1 x k log x, k N Polyharmonic Spline (PHS) =

4 Some RBFs in 1D Polyharmonic Spline RBFs Infinitely Differentiable RBFs 4

5 Example: Equi-spaced Interpolation in 1D PHS Basis Functions Approximation 5

6 Structure of the Linear System Least Squares Parabola: μ 1 + μ x + μ x 1 x 1 x 1 1 x x 1 x x 1 x 4 x 4 1 x 5 x 5 μ 1 μ μ = f 1 f f f 4 f 5 Polynomial Interpolant: μ 1 + μ x + μ x + μ 4 x + μ 5 x 4 1 x 1 x 1 x 1 4 x 1 1 x x x 4 x 1 x x x 4 x 1 x 4 x 4 x 4 4 x 4 1 x 5 x 5 x 5 4 x 5 μ 1 μ μ μ 4 μ 5 = f 1 f f f 4 f 5 5 PHS Interpolant: λ j x x j + μ1 + μ x + μ x 0 x 1 x x 1 x x 1 x 4 x 1 x 5 1 x 1 x 1 x x 1 0 x x x x 4 x x 5 1 x x x x 1 x x 0 x x 4 x x 5 1 x x x 4 x 1 x 4 x x 4 x 0 x 4 x 5 1 x 4 x 4 x 5 x 1 x 5 x x 5 x x 5 x x 5 x 5 λ 1 λ λ λ 4 λ 5 = f 1 f f f 4 f 5 j= x 1 x x x 4 x x 1 x x x 4 x μ 1 μ μ

7 Properties of Polyharmonic Splines PHS basis includes both RBFs and polynomials RBFs improve performance and allow the use of irregular nodes polynomials give convergence to smooth solutions (no saturation error) The interpolation problem is guaranteed to have a unique solution provided that polynomials are included up to the required degree, and the nodes are unisolvent. For k = 1,,, φ x = x k log x Polynomials up to degree k or higher φ x = x k+1 Polynomials up to degree k or higher Rule of thumb for modest polynomial degrees: Twice as many RBFs as polynomials Condition number of PHS A-matrix is invariant under rotation, translation, and uniform scaling No need to search for optimal shape parameter 7

8 Interpolation in D x = x, y n Given nodes x i, y i i=1 and corresponding function values f n i i=1, find a linear combination of RBF and polynomial basis functions that matches the data exactly. 1. Assume the appropriate form of the underlying approximation: n Φ x, y = λ j φ j x, y + μ k p k x, y, j=1 m k=1 where φ j x, y = φ x x j, y y j.. Require Φ to match the data at each node: Φ x i, y i = λ j φ j x i, y i n j=1 m + μ k p k x i, y i = f i, i = 1,,,, n. k=1. Enforce regularity conditions on the coefficients λ j : n λ j p k x j, y j = 0, k = 1,,,, m. j=1 4. Solve the symmetric linear system for λ j and μ k. 8

9 Differentiation Weights in D Interpolation Problem: A P T P O λ μ = f O, a ij = φ j x i, y i = φ x i x j, y i y j, i, j = 1,,,, n, p ik = p k x i, y i, i = 1,,,, n, k = 1,,,, m. Use LΦ x, y to approximate Lf x, y : LΦ n x, y = λ j Lφ j j=1 m x, y + μ k Lp k k=1 = b c λ μ = b c A P P T O weights 1 f O, b j = Lφ j x, y, j = 1,,,, n, c k = Lp k x, y, k = 1,,,, m. x, y 9

10 Derivative Approximation on the Sphere Given: nodes x, y, z on the sphere and function values f Find: differentiation matrices (DMs) W x, W y, W z to approximate x, Method: Use the fact that f x, y, zǁ is tangent to the sphere at x, y, zǁ For each node, get orthogonal unit vectors e x and e y tangent to the sphere Use D method to get matrices W x and W y that approximate x f f = e x + e f x y y f = f f x 1 = e x 1 + e f x y = e 1 y x 1 + e x y 1 y W x = diag e x 1 W x + diag e y 1 W y W y = diag e x W x + diag e y W y W z = diag e x W x + diag e y W y f and y y, z 10

11 Semi-Lagrangian Transport Governing Equations (velocity u is a known function) ρ t = u ρ ρ u, Dq Dt = t (Eulerian, short time-steps) + u q = 0. (Semi-Lagrangian, long time-steps) Quasi-Monotone Limiter for q (const. along flow trajectories) m k = min l (n) q kl and M k = max l Set q k (n+1) = min qk (n+1), Mk (n) q kl Set q k (n+1) = max qk (n+1), mk Mass Fixer tracermass σ N k=1 ρ k q k V k If tracermass < initialmass, add mass in cells with q k < M k If tracermass > initialmass, subtract mass from cells with q k > m k 11

12 Semi-Lagrangian Transport t n+1 t n 1

13 Pre-Processing Get DMs W x, W y, and W z for the continuity equation Find index of the n nearest neighbors to each node For each node x k = x k, y k, z k, write its n neighbors in terms of two orthogonal unit vectors e x, e y tangent to the sphere at x k, and one unit vector e zƹ normal to the sphere at x k, so that x kl = x kl e x + y kl e y + zƹ kl e z Ƹ, l = 1,,,, n. P k 1 Set C k = A k I n T, where P k O 6 6 O 6 n A k ij = φ x ki x kj, y ki y kj, i, j = 1,,,, n, P k = 1, x k, y k, x k, x k y k, y k. 1

14 Time Stepping 1. Step ρ t = u ρ ρ u from t n to t n+1 using several explicit Eulerian time steps (RK). Step x = u from t n+1 to t n to get departure points (RK4). Find the nearest fixed neighbor to each departure point 4. Use the corresponding pre-calculated cardinal coefficients C k and the newly formed row-vectors b k to get rows of the interpolation matrix W W k = b k C k 5. Update q on fixed nodes using weights W 6. Cycle quasi-monotone limiter and mass-fixer until the tracer mass is nearly equal to the initial tracer mass (diff<1e-1) 7. Repeat 14

15 Hyperviscosity Add a small dissipative term to the continuity equation ρ t = u ρ ρ u + γ max u Δx K 1 Δ K ρ Reduce high-frequency noise while keeping order of convergence intact Achieve stability in time using explicit time-stepping PHS are ideal for hyperviscosity, because applying the Laplace operator to a PHS returns another PHS φ x, y = x + y m/ Δφ x, y = φ x + φ y = m x + y (m )/ Parameter γ R is determined experimentally at low resolution, and remains unchanged as resolution increases 15

16 Transport Test Cases on the Sphere Initial Condition q 0 (ρ 0 = 1 in all cases) Taken from Nair and Lauritzen, 010 (NL010) Gaussian Hills (infinitely differentiable) Cosine Bells (once continuously differentiable) Slotted Cylinders (not continuous) Velocity Field (NL010) Case 1: Translating, vorticity-dominated flow CFL max = Case : Translating, divergence-dominated flow CFL max 5 Spatial Approximations (Minimum Energy (ME) Nodes) max u Δt Δx 8 Number of nodes N = 4 (576), 48 (04), , Interpolation (semi-lagrangian tracer transport) x + p1 + n19 Derivative Approximation (Eulerian continuity equation) x log x + p5 + n4 Time-stepping from t = 0 to t = 5 (one revolution) 16

17 θ Nodes and Initial Conditions for q Gaussian Hills (GH) Cosine Bells (CB) π N = 4 = 576 ME nodes Slotted Cylinders (SC) π λ 0 π 17

18 θ Time Snapshots, Velocity Case π 4 5 π λ 0 π 18

19 θ Time Snapshots, Velocity Case 0 1 π 4 5 π λ 0 π 19

20 GH, Velocity Case 1, Unlimited q q exact log 10 q exact log 10 N 0

21 CB, Velocity Case 1, Unlimited q q exact log 10 q exact log 10 N 1

22 SC, Velocity Case 1, Unlimited q q exact log 10 q exact log 10 N

23 N = θ N = 96. N = GH, Velocity Case 1 Unlimited Limited π π λ 0 π

24 N = θ N = 96. N = CB, Velocity Case 1 Unlimited Limited π π λ 0 π 4

25 N = θ N = 96. N = SC, Velocity Case 1 Unlimited Limited π π λ 0 π 5

26 N = θ N = 96. N = GH, Velocity Case Unlimited Limited π π λ 0 π 6

27 N = θ N = 96. N = CB, Velocity Case Unlimited Limited π π λ 0 π 7

28 N = θ N = 96. N = SC, Velocity Case Unlimited Limited π π λ 0 π 8

29 Advantages of Semi-Lagrangian Transport Large Time steps max u Δt Δx 1 Simple governing equation and solution algorithm Dq Dt = 0 D Dt = t + u q is constant along flow trajectories No spatial derivatives (1) time-step for departure points, () interpolate for new values of q No need for hyperviscosity High frequencies automatically damped by repeated interpolation Numerical solutions remain bounded even if the node set is poorly distributed Simple limiter to reduce oscillations and preserve bounds 9

30 Shallow Water Model Governing Equations: u t = u u f k u g h, h t = u h h u. f = Ω sin θ, where Ω = angular velocity of Earth, θ = latitude k is the unit normal to the sphere g is gravitational acceleration h = h s x, y, z + h x, y, z, t is the depth of the fluid Note: The velocity u is adjusted after every Runge-Kutta stage to remain tangent to the sphere u u u k k 0

31 Nodes Maximum Determinant (MD) Hammersley 1

32 Shallow Water Test Cases Taken from Williamson et al, JCP 199 Steady-state smooth flow h s = 0 Exact solution known Flow over an isolated mountain h s is a cone-shaped mountain centered at λ, θ = π, π 6 Exact solution unavailable Rossby-Haurwitz Wave u 0 and h 0 satisfy the barotropic vorticity equations h s = 0 Exact solution unavailable

33 Parameters for Shallow Water Tests Derivative approximations,, x y z φ x = x log x Polynomials up to degree 5 Stencil size 4 (twice as many RBFs as polynomials) Hyperviscosity Δ φ x = x 7 Polynomials up to degree 5 Stencil size 4 Parameter γ = Time Stepping ( stage, rd order Runge-Kutta) N = 4 N = 48 N = 96 N = 19 Δt (minutes)

34 Error Growth in Time MD Hammersley h h exact log 10 h exact h h exact log 10 h exact t (days) t (days) 4

35 Convergence Verification t = 5 h h exact log 10 h exact log 10 N 5

36 θ Time Snapshots, Isolated Mountain t = 0 t = π t = 6 t = 9 t = 1 t = 15 π π λ π 6

37 N = 19 θ N = 96 N = 48 Flow over Mountain, t = 15 MD Hammersley π π π λ π 7

38 N = 19 θ N = 96 N = 48 Rossby-Haurwitz, t = 14 MD Hammersley π π π λ π 8

39 Strengths of PHS RBF-FD Simple and accurate on the sphere Local and well suited for parallel computations Free from coordinate singularities Discretize directly from Cartesian equations Geometrically flexible Does not require a mesh Static Node Refinement Dynamic Node Refinement Robust Same configuration (basis, stencil-size, hyperviscosity parameter) runs on a wide variety of node-sets and test problems x log x + p5 + n4 for first derivative approximations 9

40 Future Work Transport D test cases on spherical shell (DCMIP test cases) More sophisticated fixer/limiter procedure Reduce parallel communication Shallow water equations Quantitative comparison to other methods Additional tests on the sphere from Williamson et al, JCP 199 Forced nonlinear system with a translating Low Evolution of highly nonlinear wave Nonhydrostatic Dynamical Core for climate/weather D benchmarks in Cartesian geometry with topography Fully D without using a terrain-following coordinate transformation Eulerian dynamics, semi-lagrangian transport with fixer/limiter 40

41 References Natasha Flyer, Gregory A. Barnett, Louis J. Wicker, Enhancing finite differences with radial basis functions: Experiments on the Navier Stokes equations, Journal of Computational Physics, Volume 16, pages 9-6, July 016. Natasha Flyer, Erik Lehto, Sebastien Blaise, Grady B. Wright, Amik St-Cyr, A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere, Journal of Computational Physics, Vol 1, Issue 11, pages , 01. Bengt Fornberg and Natasha Flyer, A Primer on Radial Basis Functions with Applications to the Geosciences, CBMS-NSF, Regional Conference Series in Applied Mathematics (No. 87), September 015. Armin Iske, On the Approximation Order and Numerical Stability of Local Lagrange Interpolation by Polyharmonic Splines, Modern Developments in Multivariate Approximation, International Series of Numerical Mathematics, Vol 145, 00. Peter H. Lauritzen, Christiane Jablonowski, Mark A. Taylor, Ramachandran D. Nair, Numerical Techniques for Global Atmospheric Models, Lecture notes in Computational Science and Engineering, Vol 80, Springer-Verlag Berlin Heidelberg, 011. Ramachandran D. Nair, Peter H. Lauritzen, A class of deformational flow test cases for linear transport problems on the sphere, Journal of Computational Physics, Vol 9, Issue, pages , November 010. David L. Williamson, John B. Drake, James J. Hack, Rüdiger Jakob, Paul N. Swarztrauber, A standard test set for numerical approximations to the shallow water equations in spherical geometry, Journal of Computational Physics, Vol 10, Issue 1, pages 11-4,