[PPT] - Atmospheric Dynamics with Polyharmonic Spline RBFs Greg Barnett PowerPoint Presentation

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Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly

wned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

SAND2017-8166 C

Atmospheric Dynamics with Polyharmonic Spline RBFs

Greg Barnett

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Outline

▪ Polyharmonic Spline (PHS) RBFs with Polynomials

▪ 1D Example

▪ Interpolation and Differentiation Weights

▪ Interpolation/Weights in 2D ▪ 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

2

SLIDE 3

Table of RBFs

3

𝜚 𝒚 , 𝒚 ∈ ℝ𝑒, 𝜁 ∈ ℝ Name (acronym) 1 + 𝜁 𝒚

2

Multiquadric (MQ) 1 1 + 𝜁 𝒚

2

Inverse Quadratic (IQ) 1 1 + 𝜁 𝒚

2

Inverse Multiquadric (IMQ) 𝑓− 𝜁 𝒚

2

Gaussian (GA) 𝒚 2𝑙+1 𝒚 2𝑙 log 𝒚 , 𝑙 ∈ ℕ Polyharmonic Spline (PHS)

⋅ = ⋅ 2

SLIDE 4

Some RBFs in 1D

Polyharmonic Spline RBFs Infinitely Differentiable RBFs

4

SLIDE 5

Example: Equi-spaced Interpolation in 1D

PHS Basis Functions Approximation

5

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Structure of the Linear System

6

1 𝑦1 𝑦1

2

1 𝑦2 𝑦2

2

1 𝑦3 𝑦3

2

1 𝑦4 𝑦4

2

1 𝑦5 𝑦5

2

𝜈1 𝜈2 𝜈3 = 𝑔

1

𝑔

2

𝑔

3

𝑔

4

𝑔

5

1 𝑦1 𝑦1

2

𝑦1

3

𝑦1

4

1 𝑦2 𝑦2

2

𝑦2

3

𝑦2

4

1 𝑦3 𝑦3

2

𝑦3

3

𝑦3

4

1 𝑦4 𝑦4

2

𝑦4

3

𝑦4

4

1 𝑦5 𝑦5

2

𝑦5

3

𝑦5

4

𝜈1 𝜈2 𝜈3 𝜈4 𝜈5 = 𝑔

1

𝑔

2

𝑔

3

𝑔

4

𝑔

5

𝑦1 − 𝑦2 3 𝑦1 − 𝑦3 3 𝑦1 − 𝑦4 3 𝑦1 − 𝑦5 3 1 𝑦1 𝑦1

2

𝑦2 − 𝑦1 3 𝑦2 − 𝑦3 3 𝑦2 − 𝑦4 3 𝑦2 − 𝑦5 3 1 𝑦2 𝑦2

2

𝑦3 − 𝑦1 3 𝑦3 − 𝑦2 3 𝑦3 − 𝑦4 3 𝑦3 − 𝑦5 3 1 𝑦3 𝑦3

2

𝑦4 − 𝑦1 3 𝑦4 − 𝑦2 3 𝑦4 − 𝑦3 3 𝑦4 − 𝑦5 3 1 𝑦4 𝑦4

2

𝑦5 − 𝑦1 3 𝑦5 − 𝑦2 3 𝑦5 − 𝑦3 3 𝑦5 − 𝑦4 3 1 𝑦5 𝑦5

2

1 1 1 1 1 𝑦1 𝑦2 𝑦3 𝑦4 𝑦5 𝑦1

2

𝑦2

2

𝑦3

2

𝑦4

2

𝑦5

2

𝜇1 𝜇2 𝜇3 𝜇4 𝜇5 𝜈1 𝜈2 𝜈3 = 𝑔

1

𝑔

2

𝑔

3

𝑔

4

𝑔

5

Least Squares Parabola: 𝜈1 + 𝜈2𝑦 + 𝜈3𝑦2 Polynomial Interpolant: 𝜈1 + 𝜈2𝑦 + 𝜈3𝑦2 + 𝜈4𝑦3 + 𝜈5𝑦4 PHS Interpolant: ෍

𝑘=1 5

𝜇𝑘 𝑦 − 𝑦𝑘

3 + 𝜈1 + 𝜈2𝑦 + 𝜈3𝑦2

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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 𝑙 = 1,2,3, …

▪ 𝜚 𝒚 = 𝒚 2𝑙 log 𝒚

▪ Polynomials up to degree 𝑙 or higher

▪ 𝜚 𝒚 = 𝒚 2𝑙+1

▪ Polynomials up to degree 𝑙 or higher

▪ Rule of thumb for modest polynomial degrees: Twice as many RBFs as polynomials

▪ Condition number of PHS 𝐵-matrix is invariant under rotation, translation, and uniform scaling ▪ No need to search for optimal shape parameter

7

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Interpolation in 2D 𝒚 = 𝑦, 𝑧

Given nodes 𝑦𝑗, 𝑧𝑗

𝑗=1 𝑜

and corresponding function values 𝑔

𝑗 𝑗=1 𝑜

, find a linear combination of RBF and polynomial basis functions that matches the data exactly.

1. Assume the appropriate form of the underlying approximation:

Φ 𝑦, 𝑧 = ෍

𝑘=1 𝑜

𝜇𝑘𝜚𝑘 𝑦, 𝑧 + ෍

𝑙=1 𝑛

𝜈𝑙𝑞𝑙 𝑦, 𝑧 , where 𝜚𝑘 𝑦, 𝑧 = 𝜚 𝑦 − 𝑦𝑘, 𝑧 − 𝑧𝑘 .

2. Require Φ to match the data at each node:

Φ 𝑦𝑗, 𝑧𝑗 = ෍

𝑘=1 𝑜

𝜇𝑘𝜚𝑘 𝑦𝑗, 𝑧𝑗 + ෍

𝑙=1 𝑛

𝜈𝑙𝑞𝑙 𝑦𝑗, 𝑧𝑗 = 𝑔

𝑗,

𝑗 = 1,2,3, … , 𝑜.

3. Enforce regularity conditions on the coefficients 𝜇𝑘 :

෍

𝑘=1 𝑜

𝜇𝑘𝑞𝑙 𝑦𝑘, 𝑧𝑘 = 0, 𝑙 = 1,2,3, … , 𝑛.

4. Solve the symmetric linear system for 𝜇𝑘 and 𝜈𝑙 .

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Differentiation Weights in 2D

Interpolation Problem: 𝐁 𝐐 𝐐𝑈 𝐏 𝝁 𝝂 = 𝒈 𝑷 , 𝑏𝑗𝑘 = 𝜚𝑘 𝑦𝑗, 𝑧𝑗 = 𝜚 𝑦𝑗 − 𝑦𝑘, 𝑧𝑗 − 𝑧𝑘 , 𝑗, 𝑘 = 1,2,3, … , 𝑜, 𝑞𝑗𝑙 = 𝑞𝑙 𝑦𝑗, 𝑧𝑗 , 𝑗 = 1,2,3, … , 𝑜, 𝑙 = 1,2,3, … , 𝑛. Use 𝑀Φ ෤ 𝑦, ෤ 𝑧 to approximate 𝑀𝑔 ෤ 𝑦, ෤ 𝑧 : 𝑀Φ ෤ 𝑦, ෤ 𝑧 = ෍

𝑘=1 𝑜

𝜇𝑘 𝑀𝜚𝑘 ෤ 𝑦, ෤ 𝑧 + ෍

𝑙=1 𝑛

𝜈𝑙 𝑀𝑞𝑙 ෤ 𝑦, ෤ 𝑧 = 𝒄 𝒅 𝝁 𝝂 = 𝒄 𝒅 𝐁 𝐐 𝐐𝑈 𝐏

−1 weights

𝒈 𝑷 , 𝑐

𝑘 = 𝑀𝜚𝑘

෤ 𝑦, ෤ 𝑧 , 𝑘 = 1,2,3, … , 𝑜, 𝑑𝑙 = 𝑀𝑞𝑙 ෤ 𝑦, ෤ 𝑧 , 𝑙 = 1,2,3, … , 𝑛.

9

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Derivative Approximation on the Sphere

▪ Given: nodes 𝒚, 𝒛, 𝒜 on the sphere and function values 𝒈 ▪ Find: differentiation matrices (DMs) 𝐗𝑦, 𝐗𝑧, 𝐗𝑨 to approximate

𝜖 𝜖𝑦, 𝜖 𝜖𝑧, 𝜖 𝜖𝑨

▪ Method: Use the fact that 𝛼𝑔 ෤ 𝑦, ෤ 𝑧, ǁ 𝑨 is tangent to the sphere at ෤ 𝑦, ෤ 𝑧, ǁ 𝑨

▪ For each node, get orthogonal unit vectors 𝒇 ො

𝑦 and 𝒇 ො 𝑧 tangent to the sphere

▪ Use 2D method to get matrices 𝐗ො

𝑦 and 𝐗ො 𝑧 that approximate 𝜖 𝜖 ො 𝑦 and 𝜖 𝜖 ො 𝑧

▪ 𝛼𝑔 = 𝒇 ො

𝑦 𝜖𝑔 𝜖 ො 𝑦 + 𝒇 ො 𝑧 𝜖𝑔 𝜖 ො 𝑧

▪

𝜖𝑔 𝜖𝑦 = 𝛼𝑔 1 = 𝒇 ො 𝑦 1 𝜖𝑔 𝜖 ො 𝑦 + 𝒇 ො 𝑧 1 𝜖𝑔 𝜖 ො 𝑧 =

𝒇 ො

𝑦 1 𝜖 𝜖 ො 𝑦 + 𝒇 ො 𝑧 1 𝜖 𝜖 ො 𝑧 𝑔

▪ 𝐗𝑦 = diag 𝒇 ො

𝑦 1 𝐗ො 𝑦 + diag

𝒇 ො

𝑧 1 𝐗ො 𝑧

▪ 𝐗𝑧 = diag 𝒇 ො

𝑦 2 𝐗ො 𝑦 + diag

𝒇 ො

𝑧 2 𝐗ො 𝑧

▪ 𝐗𝑨 = diag 𝒇 ො

𝑦 3 𝐗ො 𝑦 + diag

𝒇 ො

𝑧 3 𝐗ො 𝑧

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Semi-Lagrangian Transport

▪ Governing Equations (velocity 𝒗 is a known function)

▪

𝜖𝜍 𝜖𝑢 = −𝒗 ⋅ 𝛼𝜍 − 𝜍𝛼 ⋅ 𝒗,

(Eulerian, short time-steps) ▪

𝐸𝑟 𝐸𝑢 = 𝜖 𝜖𝑢 + 𝒗 ⋅ 𝛼 𝑟 = 0.

(Semi-Lagrangian, long time-steps)

▪ Quasi-Monotone Limiter for 𝑟 (const. along flow trajectories)

▪ 𝑛𝑙 = min

ℓ

𝑟𝑙ℓ

(𝑜) and 𝑁𝑙 = max ℓ

𝑟𝑙ℓ

(𝑜)

▪ Set 𝑟𝑙

(𝑜+1) = min 𝑟𝑙 (𝑜+1), 𝑁𝑙

▪ Set 𝑟𝑙

(𝑜+1) = max 𝑟𝑙 (𝑜+1), 𝑛𝑙

▪ Mass Fixer tracerMass ≡ σ𝑙=1

𝑂

𝜍𝑙𝑟𝑙𝑊

𝑙

▪ If tracerMass < initialMass, add mass in cells with 𝑟𝑙 < 𝑁𝑙 ▪ If tracerMass > initialMass, subtract mass from cells with 𝑟𝑙 > 𝑛𝑙

11

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Semi-Lagrangian Transport

12

𝑢𝑜+1 𝑢𝑜

SLIDE 13

Pre-Processing

▪ Get DMs 𝐗𝑦, 𝐗𝑧, and 𝐗𝑨 for the continuity equation ▪ Find index of the 𝑜 nearest neighbors to each node ▪ For each node 𝒚𝑙 = 𝑦𝑙, 𝑧𝑙, 𝑨𝑙 , write its 𝑜 neighbors in terms of two orthogonal unit vectors 𝒇 ො

𝑦, 𝒇 ො 𝑧 tangent to the

sphere at 𝒚𝑙, and one unit vector 𝒇 Ƹ

𝑨 normal to the sphere at

𝒚𝑙, so that 𝒚𝑙ℓ = ො 𝑦𝑙ℓ𝒇 ො

𝑦 + ො

𝑧𝑙ℓ𝒇 ො

𝑧 + Ƹ

𝑨𝑙ℓ𝒇 Ƹ

𝑨,

ℓ = 1,2,3, … , 𝑜. ▪ Set 𝐃𝑙 = 𝐁𝑙 𝐐𝑙 𝐐𝑙

𝑈

𝐏6×6

−1

𝐉𝑜 𝐏6×𝑜 , where 𝐁𝑙 𝑗𝑘 = 𝜚 ො 𝑦𝑙𝑗 − ො 𝑦𝑙𝑘, ො 𝑧𝑙𝑗 − ො 𝑧𝑙𝑘 , 𝑗, 𝑘 = 1,2,3, … , 𝑜, 𝐐𝑙 = 𝟐, ෝ 𝒚𝑙, ෝ 𝒛𝑙, ෝ 𝒚𝑙

2, ෝ

𝒚𝑙ෝ 𝒛𝑙, ෝ 𝒛𝑙

2 .

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Time Stepping

1. Step

𝜖𝜍 𝜖𝑢 = −𝒗 ⋅ 𝛼𝜍 − 𝜍𝛼 ⋅ 𝒗 from 𝑢𝑜 to 𝑢𝑜+1 using several

explicit Eulerian time steps (RK3)

2. Step 𝒚′ = −𝒗 from 𝑢𝑜+1 to 𝑢𝑜 to get departure points (RK4)
3. Find the nearest fixed neighbor to each departure point
4. Use the corresponding pre-calculated cardinal coefficients

𝐃𝑙 and the newly formed row-vectors 𝒄𝑙 to get rows of the interpolation matrix 𝐗 𝐗𝑙⋅ = 𝒄𝑙𝐃𝑙

5. Update 𝑟 on fixed nodes using weights 𝐗
6. Cycle quasi-monotone limiter and mass-fixer until the tracer

mass is nearly equal to the initial tracer mass (diff<1e-13)

7. Repeat

14

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Hyperviscosity

▪ Add a small dissipative term to the continuity equation

▪

𝜖𝜍 𝜖𝑢 = −𝒗 ⋅ 𝛼𝜍 − 𝜍𝛼 ⋅ 𝒗 + 𝛿 max 𝒗

Δ𝑦 2𝐿−1Δ𝐿𝜍

▪ 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

perator to a PHS returns another PHS

▪ 𝜚 𝑦, 𝑧 = 𝑦2 + 𝑧2 𝑛/2 ▪ Δ𝜚 𝑦, 𝑧 =

𝜖2𝜚 𝜖𝑦2 + 𝜖2𝜚 𝜖𝑧2 = 𝑛2 𝑦2 + 𝑧2 (𝑛−2)/2

▪ Parameter 𝛿 ∈ ℝ is determined experimentally at low resolution, and remains unchanged as resolution increases

15

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Transport Test Cases on the Sphere

▪ Initial Condition 𝑟0 (𝜍0 = 1 in all cases)

▪ Taken from Nair and Lauritzen, 2010 (NL2010)

▪ Gaussian Hills (infinitely differentiable) ▪ Cosine Bells (once continuously differentiable) ▪ Slotted Cylinders (not continuous)

▪ Velocity Field (NL2010)

▪ Case 1: Translating, vorticity-dominated flow CFLmax = max 𝒗 Δ𝑢

Δ𝑦

≈ 8 ▪ Case 2: Translating, divergence-dominated flow CFLmax ≈ 5

▪ Spatial Approximations (Minimum Energy (ME) Nodes)

▪ Number of nodes 𝑂 = 242(576), 482(2304), 962 9216 , 1922 36864 ▪ Interpolation (semi-Lagrangian tracer transport)

▪ 𝒚 3 + 𝑞1 + 𝑜19

▪ Derivative Approximation (Eulerian continuity equation)

▪ 𝒚 2 log 𝒚 + 𝑞5 + 𝑜42

▪ Time-stepping from 𝑢 = 0 to 𝑢 = 5 (one revolution)

16

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Nodes and Initial Conditions for 𝑟

17

Gaussian Hills (GH) Cosine Bells (CB) Slotted Cylinders (SC) 𝑂 = 242 = 576 ME nodes

𝜇 𝜄 2𝜌 − 𝜌 2 𝜌 2

SLIDE 18

Time Snapshots, Velocity Case 1

18

1 2 3 4 5

𝜇 𝜄 2𝜌 − 𝜌 2 𝜌 2

SLIDE 19

▪ Large Time steps

max 𝒗 Δ𝑢 Δ𝑦

≫ 1 ▪ Simple governing equation and solution algorithm

▪

𝐸𝑟 𝐸𝑢 = 0 𝐸 𝐸𝑢 = 𝜖 𝜖𝑢 + 𝒗 ⋅ 𝛼

▪ 𝑟 is constant along flow trajectories ▪ No spatial derivatives

▪ (1) time-step for departure points, (2) interpolate for new values of 𝑟

▪ 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

29

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Shallow Water Model

Governing Equations:

𝜖𝒗 𝜖𝑢 = −𝒗 ⋅ 𝛼𝒗 − 𝑔 ෡

𝒍 × 𝒗 − 𝑕𝛼ℎ,

𝜖ℎ∗ 𝜖𝑢 = −𝒗 ⋅ 𝛼ℎ∗ − ℎ∗ 𝛼 ⋅ 𝒗 .

▪ 𝑔 = 2Ω sin 𝜄, where Ω = angular velocity of Earth, 𝜄 = latitude ▪ ෡ 𝒍 is the unit normal to the sphere ▪ 𝑕 is gravitational acceleration ▪ ℎ = ℎ𝑡 𝑦, 𝑧, 𝑨 + ℎ∗ 𝑦, 𝑧, 𝑨, 𝑢 is the depth of the fluid Note: The velocity 𝒗 is adjusted after every Runge-Kutta stage to remain tangent to the sphere 𝒗 ← 𝒗 − 𝒗 ⋅ ෡ 𝒍 ෡ 𝒍

30

SLIDE 31

Nodes

Maximum Determinant (MD) Hammersley

31

SLIDE 32

Shallow Water Test Cases

▪ Taken from Williamson et al, JCP 1992

▪ Steady-state smooth flow

▪ ℎ𝑡 = 0 ▪ Exact solution known

▪ Flow over an isolated mountain

▪ ℎ𝑡 is a cone-shaped mountain centered at 𝜇, 𝜄 = −

𝜌 2 , 𝜌 6

▪ Exact solution unavailable

▪ Rossby-Haurwitz Wave

▪ 𝒗0 and ℎ0 satisfy the barotropic vorticity equations

▪ ℎ𝑡 = 0 ▪ Exact solution unavailable

32

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Parameters for Shallow Water Tests

▪ Derivative approximations

𝜖 𝜖𝑦 , 𝜖 𝜖𝑧 , 𝜖 𝜖𝑨

▪ 𝜚 𝑦 = 𝑦

2 log 𝑦

▪ Polynomials up to degree 5 ▪ Stencil size 42 (twice as many RBFs as polynomials)

▪ Hyperviscosity Δ3

▪ 𝜚 𝑦 = 𝑦

7

▪ Polynomials up to degree 5 ▪ Stencil size 42 ▪ Parameter 𝛿 = 2−12 ≈ 2.4 × 10−4

▪ Time Stepping (3 stage, 3rd order Runge-Kutta)

33 𝑶 = 𝟑𝟓𝟑 𝑶 = 𝟓𝟗𝟑 𝑶 = 𝟘𝟕𝟑 𝑶 = 𝟐𝟘𝟑𝟑 Δ𝑢 (minutes) 36 18 9 4.5

SLIDE 34

Error Growth in Time

MD Hammersley

34 log10 ℎ − ℎ𝑓𝑦𝑏𝑑𝑢

2

ℎ𝑓𝑦𝑏𝑑𝑢

2

𝑢 (days) 𝑢 (days) log10 ℎ − ℎ𝑓𝑦𝑏𝑑𝑢

2

ℎ𝑓𝑦𝑏𝑑𝑢

2

SLIDE 35

Convergence Verification 𝑢 = 5

35 log10 𝑂 log10 ℎ − ℎ𝑓𝑦𝑏𝑑𝑢

2

ℎ𝑓𝑦𝑏𝑑𝑢

2

SLIDE 36

Time Snapshots, Isolated Mountain

36

𝑢 = 0 𝑢 = 3 𝑢 = 6 𝑢 = 9 𝑢 = 12 𝑢 = 15

𝜇 𝜄 −𝜌 𝜌 − 𝜌 2 𝜌 2

SLIDE 37

Flow over Mountain, 𝑢 = 15

37

Hammersley MD 𝑂 = 482 𝑂 = 962 𝑂 = 1922

𝜇 𝜄 −𝜌 𝜌 − 𝜌 2 𝜌 2

SLIDE 38

Rossby-Haurwitz, 𝑢 = 14

38

𝑂 = 482 𝑂 = 962 𝑂 = 1922

𝜇 𝜄 −𝜌 𝜌 − 𝜌 2 𝜌 2

Hammersley MD

SLIDE 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

n a wide variety of node-sets and test problems

▪ 𝒚 2 log 𝒚 + p5 + 𝑜42 for first derivative approximations

39

SLIDE 40

Future Work

▪ Transport

▪ 3D 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 1992

▪ Forced nonlinear system with a translating Low ▪ Evolution of highly nonlinear wave

▪ Nonhydrostatic Dynamical Core for climate/weather

▪ 2D benchmarks in Cartesian geometry with topography ▪ Fully 3D without using a terrain-following coordinate transformation ▪ Eulerian dynamics, semi-Lagrangian transport with fixer/limiter

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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 316, pages 39-62, July 2016. 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 231, Issue 11, pages 4078-4095, 2012. 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 2015. 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, 2003. 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, 2011. 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 229, Issue 23, pages 8868-8887, November 2010. 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 102, Issue 1, pages 211-224, 1992.

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