POD-DEIM Based Model Order Reduction for the Spherical Shallow Water Equations with Turkel-Zwas Finite Difference Discretization

Algorithm 1

DEIM pseudo-algorithm for interpolation indices [18, 27].

INPUT: {𝑢_𝑙}^𝑚_𝑙=1⊂ℝ^𝑛 (linearly independent),

OUTPUT: 𝜌=[𝜌₁,𝜌₂,…,𝜌_𝑚]∈ℕ^𝑚,

Step  1. [|𝜓|𝜌₁]=max|𝑢₁|, 𝜓∈ℝ and 𝜌₁ is the component position of the

   largest absolute value of 𝑢₁, with the smallest index taken in case of a tie,

Step  2. 𝑈=[𝑢₁]∈ℝ^𝑛, 𝑃=[𝑒_𝜌1]∈ℝ^𝑛, 𝜌=[𝜌₁]∈ℕ⁺,

Step  3. For 𝑙=2,…,𝑚 do

  (a) Solve (𝑃^𝑇𝑈)𝑐=𝑃^𝑇𝑢_𝑙 for 𝑐∈ℝ^𝑙−1, 𝑈, 𝑃∈ℝ^{𝑛×(𝑙−1)},

  (b) 𝑟=𝑢_𝑙−𝑈𝑐, 𝑟∈ℝ^𝑛,

  (c) |𝜓|𝜌_𝑙=max{|𝑟|},

  (d) 𝑈←𝑈𝑢_𝑙, 𝑃←[𝑃𝑒_𝜌𝑙], 𝜌𝜌𝜌←_𝑙,

  end For.