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SAND2017-8166 C

Atmospheric Dynamics with Polyharmonic Spline RBFs

Greg Barnett

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

Table of RBFs

3

𝜙 𝒙 , 𝒙 ∈ ℝ𝑑 , 𝜀 ∈ ℝ Name (acronym)

1 + 𝜀 𝒙 2 Multiquadric (MQ)

1

1 + 𝜀 𝒙 2

Inverse Quadratic (IQ)

1

1 + 𝜀 𝒙 2

Inverse Multiquadric (IMQ)

𝑒− 𝜀 𝒙 2Gaussian (GA)

𝒙 2𝑘+1

𝒙 2𝑘 log 𝒙 , 𝑘 ∈ ℕPolyharmonic Spline (PHS)

⋅ = ⋅ 2

Some RBFs in 1D

Polyharmonic Spline RBFs Infinitely Differentiable RBFs

4

Example: Equi-spaced Interpolation in 1D

PHS Basis Functions Approximation

5

Structure of the Linear System

6

1 𝑥1 𝑥12

1 𝑥2 𝑥22

1 𝑥3 𝑥32

1 𝑥4 𝑥42

1 𝑥5 𝑥52

𝜇1𝜇2𝜇3

=

𝑓1𝑓2𝑓3𝑓4𝑓5

1 𝑥1 𝑥12 𝑥1

3 𝑥14

1 𝑥2 𝑥22 𝑥2

3 𝑥24

1 𝑥3 𝑥32 𝑥3

3 𝑥34

1 𝑥4 𝑥42 𝑥4

3 𝑥44

1 𝑥5 𝑥52 𝑥5

3 𝑥54

𝜇1𝜇2𝜇3𝜇4𝜇5

=

𝑓1𝑓2𝑓3𝑓4𝑓5

0 𝑥1 − 𝑥23 𝑥1 − 𝑥3

3 𝑥1 − 𝑥43 𝑥1 − 𝑥5

3 1 𝑥1 𝑥12

𝑥2 − 𝑥13 0 𝑥2 − 𝑥3

3 𝑥2 − 𝑥43 𝑥2 − 𝑥5

3 1 𝑥2 𝑥22

𝑥3 − 𝑥13 𝑥3 − 𝑥2

3 0 𝑥3 − 𝑥43 𝑥3 − 𝑥5

3 1 𝑥3 𝑥32

𝑥4 − 𝑥13 𝑥4 − 𝑥2

3 𝑥4 − 𝑥33 0 𝑥4 − 𝑥5

3 1 𝑥4 𝑥42

𝑥5 − 𝑥13 𝑥5 − 𝑥2

3 𝑥5 − 𝑥33 𝑥5 − 𝑥4

3 0 1 𝑥5 𝑥52

1 1 1 1 1 0 0 0𝑥1 𝑥2 𝑥3 𝑥4 𝑥5 0 0 0

𝑥12 𝑥2

2 𝑥32 𝑥4

2 𝑥52 0 0 0

𝜆1𝜆2𝜆3𝜆4𝜆5

𝜇1𝜇2𝜇3

=

𝑓1𝑓2𝑓3𝑓4𝑓5

000

Least Squares Parabola:𝜇1 + 𝜇2𝑥 + 𝜇3𝑥

2

Polynomial Interpolant:𝜇1 + 𝜇2𝑥 + 𝜇3𝑥

2 + 𝜇4𝑥3 + 𝜇5𝑥

4

PHS Interpolant:

𝑗=1

5

𝜆𝑗 𝑥 − 𝑥𝑗3+ 𝜇1 + 𝜇2𝑥 + 𝜇3𝑥

2

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 = 1,2,3, …▪ 𝜙 𝒙 = 𝒙 2𝑘 log 𝒙

▪ Polynomials up to degree 𝑘 or higher

▪ 𝜙 𝒙 = 𝒙 2𝑘+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

Interpolation in 2D 𝒙 = 𝑥, 𝑦

Given nodes 𝑥𝑖 , 𝑦𝑖 𝑖=1𝑛 and corresponding function values 𝑓𝑖 𝑖=1

𝑛 , 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 𝑥, 𝑦 =

𝑗=1

𝑛

𝜆𝑗𝜙𝑗 𝑥, 𝑦 +

𝑘=1

𝑚

𝜇𝑘𝑝𝑘 𝑥, 𝑦 ,

where 𝜙𝑗 𝑥, 𝑦 = 𝜙 𝑥 − 𝑥𝑗 , 𝑦 − 𝑦𝑗 .

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

Φ 𝑥𝑖 , 𝑦𝑖 =

𝑗=1

𝑛

𝜆𝑗𝜙𝑗 𝑥𝑖 , 𝑦𝑖 +

𝑘=1

𝑚

𝜇𝑘𝑝𝑘 𝑥𝑖 , 𝑦𝑖 = 𝑓𝑖 , 𝑖 = 1,2,3, … , 𝑛.

3. Enforce regularity conditions on the coefficients 𝜆𝑗 :

𝑗=1

𝑛

𝜆𝑗𝑝𝑘 𝑥𝑗, 𝑦𝑗 = 0, 𝑘 = 1,2,3,… ,𝑚.

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

8

Differentiation Weights in 2D

Interpolation Problem:𝐀 𝐏𝐏𝑇 𝐎

𝝀𝝁

=𝒇𝑶

,

𝑎𝑖𝑗 = 𝜙𝑗 𝑥𝑖 , 𝑦𝑖 = 𝜙 𝑥𝑖 − 𝑥𝑗 , 𝑦𝑖 − 𝑦𝑗 , 𝑖, 𝑗 = 1,2,3, … , 𝑛,

𝑝𝑖𝑘 = 𝑝𝑘 𝑥𝑖 , 𝑦𝑖 , 𝑖 = 1,2,3, … , 𝑛, 𝑘 = 1,2,3, … ,𝑚.

Use 𝐿Φ 𝑥, 𝑦 to approximate 𝐿𝑓 𝑥, 𝑦 :

𝐿Φ 𝑥, 𝑦 =

𝑗=1

𝑛

𝜆𝑗 𝐿𝜙𝑗 𝑥, 𝑦 +

𝑘=1

𝑚

𝜇𝑘 𝐿𝑝𝑘 𝑥, 𝑦

= 𝒃 𝒄𝝀𝝁

= 𝒃 𝒄𝐀 𝐏𝐏𝑇 𝐎

−1

weights

𝒇𝑶

,

𝑏𝑗 = 𝐿𝜙𝑗 𝑥, 𝑦 , 𝑗 = 1,2,3, … , 𝑛,

𝑐𝑘 = 𝐿𝑝𝑘 𝑥, 𝑦 , 𝑘 = 1,2,3, … ,𝑚.

9

Derivative Approximation on the Sphere

▪ Given: nodes 𝒙, 𝒚, 𝒛 on the sphere and function values 𝒇

▪ Find: differentiation matrices (DMs) 𝐖𝑥, 𝐖𝑦, 𝐖𝑧 to approximate 𝜕

𝜕𝑥, 𝜕

𝜕𝑦, 𝜕

𝜕𝑧

▪ Method: Use the fact that 𝛻𝑓 𝑥, 𝑦, ǁ𝑧 is tangent to the sphere at 𝑥, 𝑦, ǁ𝑧

▪ For each node, get orthogonal unit vectors 𝒆 ො𝑥 and 𝒆 ො𝑦 tangent to the sphere

▪ Use 2D method to get matrices 𝐖ො𝑥 and 𝐖ො𝑦 that approximate 𝜕

𝜕 ො𝑥and

𝜕

𝜕 ො𝑦

▪ 𝛻𝑓 = 𝒆 ො𝑥𝜕𝑓

𝜕 ො𝑥+ 𝒆 ො𝑦

𝜕𝑓

𝜕 ො𝑦

▪𝜕𝑓

𝜕𝑥= 𝛻𝑓 1 = 𝒆 ො𝑥 1

𝜕𝑓

𝜕 ො𝑥+ 𝒆 ො𝑦 1

𝜕𝑓

𝜕 ො𝑦= 𝒆 ො𝑥 1

𝜕

𝜕 ො𝑥+ 𝒆 ො𝑦 1

𝜕

𝜕 ො𝑦𝑓

▪ 𝐖𝑥 = diag 𝒆 ො𝑥 1 𝐖ො𝑥 + diag 𝒆 ො𝑦 1𝐖ො𝑦

▪ 𝐖𝑦 = diag 𝒆 ො𝑥 2 𝐖ො𝑥 + diag 𝒆 ො𝑦 2𝐖ො𝑦

▪ 𝐖𝑧 = diag 𝒆 ො𝑥 3 𝐖ො𝑥 + diag 𝒆 ො𝑦 3𝐖ො𝑦

10

Semi-Lagrangian Transport

▪ Governing Equations (velocity 𝒖 is a known function)

▪𝜕𝜌

𝜕𝑡= −𝒖 ⋅ 𝛻𝜌 − 𝜌𝛻 ⋅ 𝒖, (Eulerian, short time-steps)

▪𝐷𝑞

𝐷𝑡=

𝜕

𝜕𝑡+ 𝒖 ⋅ 𝛻 𝑞 = 0. (Semi-Lagrangian, long time-steps)

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

▪ 𝑚𝑘 = minℓ

𝑞𝑘ℓ(𝑛)

and 𝑀𝑘 = maxℓ

𝑞𝑘ℓ(𝑛)

▪ Set 𝑞𝑘(𝑛+1)

= min 𝑞𝑘(𝑛+1)

, 𝑀𝑘

▪ Set 𝑞𝑘(𝑛+1)

= max 𝑞𝑘(𝑛+1)

, 𝑚𝑘

▪ Mass Fixer tracerMass ≡ σ𝑘=1𝑁 𝜌𝑘𝑞𝑘𝑉𝑘

▪ If tracerMass < initialMass, add mass in cells with 𝑞𝑘 < 𝑀𝑘

▪ If tracerMass > initialMass, subtract mass from cells with 𝑞𝑘 > 𝑚𝑘

11

Semi-Lagrangian Transport

12

𝑡𝑛+1

𝑡𝑛

Pre-Processing

▪ Get DMs 𝐖𝑥, 𝐖𝑦, and 𝐖𝑧 for the continuity equation

▪ Find index of the 𝑛 nearest neighbors to each node

▪ For each node 𝒙𝑘 = 𝑥𝑘 , 𝑦𝑘 , 𝑧𝑘 , write its 𝑛 neighbors in

terms of two orthogonal unit vectors 𝒆 ො𝑥, 𝒆 ො𝑦 tangent to the

sphere at 𝒙𝑘, and one unit vector 𝒆 Ƹ𝑧 normal to the sphere at 𝒙𝑘, so that

𝒙𝑘ℓ = ො𝑥𝑘ℓ𝒆 ො𝑥 + ො𝑦𝑘ℓ𝒆 ො𝑦 + Ƹ𝑧𝑘ℓ𝒆 Ƹ𝑧, ℓ = 1,2,3,… , 𝑛.

▪ Set 𝐂𝑘 =𝐀𝑘 𝐏𝑘𝐏𝑘𝑇 𝐎6×6

−1𝐈𝑛

𝐎6×𝑛, where

𝐀𝑘 𝑖𝑗 = 𝜙 ො𝑥𝑘𝑖 − ො𝑥𝑘𝑗 , ො𝑦𝑘𝑖 − ො𝑦𝑘𝑗 , 𝑖, 𝑗 = 1,2,3,… , 𝑛,

𝐏𝑘 = 𝟏, ෝ𝒙𝑘 , ෝ𝒚𝑘 , ෝ𝒙𝑘2 , ෝ𝒙𝑘ෝ𝒚𝑘 , ෝ𝒚𝑘

2 .13

Time Stepping

1. Step 𝜕𝜌

𝜕𝑡= −𝒖 ⋅ 𝛻𝜌 − 𝜌𝛻 ⋅ 𝒖 from 𝑡𝑛 to 𝑡𝑛+1 using several

explicit Eulerian time steps (RK3)

2. Step 𝒙′ = −𝒖 from 𝑡𝑛+1 to 𝑡𝑛 to get departure points (RK4)

3. Find the nearest fixed neighbor to each departure point

4. Use the corresponding pre-calculated cardinal coefficients 𝐂𝑘 and the newly formed row-vectors 𝒃𝑘 to get rows of

the interpolation matrix 𝐖 𝐖𝑘⋅ = 𝒃𝑘𝐂𝑘5. 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-13)

7. Repeat

14

Hyperviscosity

▪ Add a small dissipative term to the continuity equation

▪𝜕𝜌

𝜕𝑡= −𝒖 ⋅ 𝛻𝜌 − 𝜌𝛻 ⋅ 𝒖 + 𝛾max 𝒖 Δ𝑥 2𝐾−1Δ𝐾𝜌

▪ 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 = 𝑥2 + 𝑦2 𝑚/2

▪ Δ𝜙 𝑥, 𝑦 =𝜕2𝜙

𝜕𝑥2+

𝜕2𝜙

𝜕𝑦2= 𝑚2 𝑥2 + 𝑦2 (𝑚−2)/2

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

15

Transport Test Cases on the Sphere

▪ Initial Condition 𝑞0 (𝜌0 = 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 𝒖 Δ𝑡

Δ𝑥≈ 8

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

▪ Spatial Approximations (Minimum Energy (ME) Nodes)

▪ Number of nodes 𝑁 = 242(576), 482(2304), 962 9216 , 1922 36864

▪ Interpolation (semi-Lagrangian tracer transport)

▪ 𝒙 3 + 𝑝1 + 𝑛19

▪ Derivative Approximation (Eulerian continuity equation)

▪ 𝒙 2 log 𝒙 + 𝑝5 + 𝑛42

▪ Time-stepping from 𝑡 = 0 to 𝑡 = 5 (one revolution)16

Nodes and Initial Conditions for 𝑞

17

Gaussian Hills

(GH)

Cosine Bells

(CB)

Slotted Cylinders

(SC)

𝑁 = 242 = 576ME nodes

𝜆

𝜃

0 2𝜋

−𝜋

2

𝜋

2

Time Snapshots, Velocity Case 1

18

0 1

2 3

4 5

𝜆

𝜃

0 2𝜋

−𝜋

2

𝜋

2

Time Snapshots, Velocity Case 2

19

0 1

2 3

4 5

𝜆

𝜃

0 2𝜋

−𝜋

2

𝜋

2

GH, Velocity Case 1, Unlimited

20log10 𝑁

log10

𝑞−𝑞𝑒𝑥𝑎𝑐𝑡

2

𝑞𝑒𝑥𝑎𝑐𝑡

2

CB, Velocity Case 1, Unlimited

21log10 𝑁

log10

𝑞−𝑞𝑒𝑥𝑎𝑐𝑡

2

𝑞𝑒𝑥𝑎𝑐𝑡

2

SC, Velocity Case 1, Unlimited

22log10 𝑁

log10

𝑞−𝑞𝑒𝑥𝑎𝑐𝑡

2

𝑞𝑒𝑥𝑎𝑐𝑡

2

GH, Velocity Case 1

23

𝑁=482

≈4.4

°𝑁=962

≈2.2

°𝑁=1922

≈1.1

°

𝜆

𝜃Unlimited Limited

0 2𝜋

−𝜋

2

𝜋

2

CB, Velocity Case 1

24

𝑁=482

≈4.4

°𝑁=962

≈2.2

°𝑁=1922

≈1.1

°

𝜆

𝜃Unlimited Limited

0 2𝜋

−𝜋

2

𝜋

2

SC, Velocity Case 1

25

𝑁=482

≈4.4

°𝑁=962

≈2.2

°𝑁=1922

≈1.1

°

𝜆

𝜃Unlimited Limited

0 2𝜋

−𝜋

2

𝜋

2

GH, Velocity Case 2

26

𝑁=482

≈4.4

°𝑁=962

≈2.2

°𝑁=1922

≈1.1

°

𝜆

𝜃Unlimited Limited

0 2𝜋

−𝜋

2

𝜋

2

CB, Velocity Case 2

27

𝑁=482

≈4.4

°𝑁=962

≈2.2

°𝑁=1922

≈1.1

°

𝜆

𝜃Unlimited Limited

0 2𝜋

−𝜋

2

𝜋

2

SC, Velocity Case 2

28

𝑁=482

≈4.4

°𝑁=962

≈2.2

°𝑁=1922

≈1.1

°

𝜆

𝜃Unlimited Limited

0 2𝜋

−𝜋

2

𝜋

2

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, (2) 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

29

Shallow Water Model

Governing Equations:

𝜕𝒖

𝜕𝑡= −𝒖 ⋅ 𝛻𝒖 − 𝑓 𝒌 × 𝒖 − 𝑔𝛻ℎ,

𝜕ℎ∗

𝜕𝑡= −𝒖 ⋅ 𝛻ℎ∗ − ℎ∗ 𝛻 ⋅ 𝒖 .

▪ 𝑓 = 2Ω sin 𝜃, where Ω = angular velocity of Earth, 𝜃 = latitude

▪ 𝒌 is the unit normal to the sphere

▪ 𝑔 is gravitational acceleration

▪ ℎ = ℎ𝑠 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

30

Nodes

Maximum Determinant (MD) Hammersley

31

Shallow Water Test Cases

▪ Taken from Williamson et al, JCP 1992

▪ Steady-state smooth flow▪ ℎ𝑠 = 0

▪ Exact solution known

▪ Flow over an isolated mountain

▪ ℎ𝑠 is a cone-shaped mountain centered at 𝜆, 𝜃 = −𝜋

2,𝜋

6

▪ Exact solution unavailable

▪ Rossby-Haurwitz Wave

▪ 𝒖0 and ℎ0 satisfy the barotropic vorticity equations

▪ ℎ𝑠 = 0

▪ Exact solution unavailable

32

Parameters for Shallow Water Tests

▪ Derivative approximations 𝜕

𝜕𝑥,𝜕

𝜕𝑦,𝜕

𝜕𝑧

▪ 𝜙 𝑥 = 𝑥2log 𝑥

▪ Polynomials up to degree 5

▪ Stencil size 42 (twice as many RBFs as polynomials)

▪ Hyperviscosity Δ3

▪ 𝜙 𝑥 = 𝑥7

▪ Polynomials up to degree 5

▪ Stencil size 42

▪ Parameter 𝛾 = 2−12 ≈ 2.4 × 10−4

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

33

𝑵 = 𝟐𝟒𝟐 𝑵 = 𝟒𝟖𝟐 𝑵 = 𝟗𝟔𝟐 𝑵 = 𝟏𝟗𝟐𝟐

Δ𝑡 (minutes) 36 18 9 4.5

Error Growth in Time

MD Hammersley

34

log10

ℎ−ℎ𝑒𝑥𝑎𝑐𝑡

2

ℎ𝑒𝑥𝑎𝑐𝑡

2

𝑡 (days) 𝑡 (days)

log10

ℎ−ℎ𝑒𝑥𝑎𝑐𝑡

2

ℎ𝑒𝑥𝑎𝑐𝑡

2

Convergence Verification 𝑡 = 5

35log10 𝑁

log10

ℎ−ℎ𝑒𝑥𝑎𝑐𝑡

2

ℎ𝑒𝑥𝑎𝑐𝑡

2

Time Snapshots, Isolated Mountain

36

𝑡 = 0 𝑡 = 3

𝑡 = 6 𝑡 = 9

𝑡 = 12 𝑡 = 15

𝜆

𝜃

−𝜋 𝜋

−𝜋

2

𝜋

2

Flow over Mountain, 𝑡 = 15

37

HammersleyMD

𝑁=482

𝑁=962

𝑁=1922

𝜆

𝜃

−𝜋 𝜋

−𝜋

2

𝜋

2

Rossby-Haurwitz, 𝑡 = 14

38

𝑁=482

𝑁=962

𝑁=1922

𝜆

𝜃

−𝜋 𝜋

−𝜋

2

𝜋

2

HammersleyMD

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 2 log 𝒙 + p5 + 𝑛42 for first derivative approximations

39

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/limiter40

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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