Boundary Conditions

Source

Model

(S, T)::Form0
              
(Ṡ, T_up)::Form0
              
v::Form1
              
v_up::Form1
              
Ṫ == ∂ₜ(T)
              
Ṡ == ∂ₜ(S)
              
v̇ == ∂ₜ(v)
              
Ṫ == ∂_spatial(T_up)
              
v̇ == ∂_noslip(v_up)

Energy balance

Source

energy balance equation from Budyko Sellers

Model

(Tₛ, ASR, OLR, HT)::Form0
              
C::Constant
              
Tₛ̇ == ∂ₜ(Tₛ)
              
Tₛ̇ == ((ASR - OLR) + HT) ./ C

Equation of State

Source

Model

(b, T, S)::Form0
              
(g, α, β)::Constant
              
b == g * (α * T - β * S)

Glens Law

Source

Nye, J. F. (1957). The Distribution of Stress and Velocity in Glaciers and Ice-Sheets. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 239(1216), 113–133. http://www.jstor.org/stable/100184

Model

Γ::Form1
              
(A, ρ, g, n)::Constant
              
Γ == (2 / (n + 2)) * A * (ρ * g) ^ n

Halfar (Eq. 2)

Source

Halfar, P. (1981), On the dynamics of the ice sheets, J. Geophys. Res., 86(C11), 11065–11072, doi:10.1029/JC086iC11p11065

Model

h::Form0
              
Γ::Form1
              
n::Constant
              
∂ₜ(h) == (∘(⋆, d, ⋆))(((Γ * d(h)) ∧ mag(♯(d(h))) ^ (n - 1)) ∧ h ^ (n + 2))

Insolation

Source

Model

Q::Form0
              
cosϕᵖ::Constant
              
Q == 450cosϕᵖ

Tracer

Source

Model

(c, C, F, c_up)::Form0
              
(v, V, q)::Form1
              
c_up == (((-1 * (⋆)(L(v, (⋆)(c))) - (⋆)(L(V, (⋆)(c)))) - (⋆)(L(v, (⋆)(C)))) - (∘(⋆, d, ⋆))(q)) + F

Warming

Source

Model

Tₛ::Form0
              
A::Form1
              
A == avg₀₁(5.8282 * 10 ^ (-0.236Tₛ) * 1.65e7)

Gompertz

Source

Eq. 6 from Yi et al. A Review of Mathematical Models for Tumor Dynamics and Treatment Resistance Evolution of Solid Tumors

Model

(C, fC)::Form0
              
Cmax::Constant
              
fC == C * ln(Cmax / C)

TumorInvasion

Source

Eq. 35 from Yi et al. A Review of Mathematical Models for Tumor Dynamics and Treatment Resistance Evolution of Solid Tumors

Model

(C, fC)::Form0
              
(Dif, Kd, Cmax)::Constant
              
∂ₜ(C) == (Dif * Δ(C) + fC) - Kd * C

Logistic

Source

Eq. 5 from Yi et al. A Review of Mathematical Models for Tumor Dynamics and Treatment Resistance Evolution of Solid Tumors

Model

(C, fC)::Form0
              
Cmax::Constant
              
fC == C * (1 - C / Cmax)

Kealy

Source

Model

(n, w)::DualForm0
              
dX::Form1
              
(a, ν)::Constant
              
∂ₜ(w) == ((a - w) - w * n ^ 2) + ν * Δ(w)

Klausmeier (Eq. 2a)

Source

Klausmeier, CA. “Regular and irregular patterns in semiarid vegetation.” Science (New York, N.Y.) vol. 284,5421 (1999): 1826-8. doi:10.1126/science.284.5421.1826

Model

(n, w)::DualForm0
              
dX::Form1
              
(a, ν)::Constant
              
∂ₜ(w) == ((a - w) - w * n ^ 2) + ν * ℒ(dX, w)

Klausmeier (Eq. 2b)

Source

ibid.

Model

(n, w)::DualForm0
              
m::Constant
              
∂ₜ(n) == (w * n ^ 2 - m * n) + Δ(n)

Lejeune

Source

Lejeune, O., & Tlidi, M. (1999). A Model for the Explanation of Vegetation Stripes (Tiger Bush). Journal of Vegetation Science, 10(2), 201–208. https://doi.org/10.2307/3237141

Model

ρ::Form0
              
(μ, Λ, L)::Constant
              
∂ₜ(ρ) == (ρ * (((1 - μ) + (Λ - 1) * ρ) - ρ * ρ) + 0.5 * (L * L - ρ) * Δ(ρ)) - 0.125 * ρ * Δ(ρ) * Δ(ρ)

Turing Continuous Ring

Source

Model

(X, Y)::Form0
              
(μ, ν, a, b, c, d)::Constant
              
∂ₜ(X) == a * X + b * Y + μ * Δ(X)
              
∂ₜ(Y) == c * X + d * Y + ν * Δ(X)

Heat Transfer

Source

Model

(HT, Tₛ)::Form0
              
(D, cosϕᵖ, cosϕᵈ)::Constant
              
HT == (D ./ cosϕᵖ) .* (⋆)(d(cosϕᵈ .* (⋆)(d(Tₛ))))

Outgoing Longwave Radiation

Source

Model

(Tₛ, OLR)::Form0
              
(A, B)::Constant
              
OLR == A .+ B .* Tₛ

Absorbed Shortwave Radiation

Source

The proportion of light reflected by a surface is the albedo. The absorbed shortwave radiation is the complement of this quantity.

Model

(Q, ASR)::Form0
              
α::Constant
              
ASR == (1 .- α) .* Q

Advection

Source

Advection refers to the transport of a bulk along a vector field.

Model

C::Form0
              
(ϕ, V)::Form1
              
ϕ == C ∧₀₁ V

Ficks Law

Source

Equation for diffusion first stated by Adolf Fick. The diffusion flux is proportional to the concentration gradient.

Model

C::Form0
              
ϕ::Form1
              
ϕ == k(d₀(C))

IceBlockingWater

Source

Model

h::Form0
              
(𝐮, w)::DualForm1
              
w == (1 - σ(h)) ∧ᵖᵈ₀₁ 𝐮

Jordan-Kinderlehrer-Otto

Source

Jordan, R., Kinderlehrer, D., & Otto, F. (1998). The Variational Formulation of the Fokker–Planck Equation. In SIAM Journal on Mathematical Analysis (Vol. 29, Issue 1, pp. 1–17). Society for Industrial & Applied Mathematics (SIAM). https://doi.org/10.1137/s0036141096303359

Model

(ρ, Ψ)::Form0
              
β⁻¹::Constant
              
∂ₜ(ρ) == (∘(⋆, d, ⋆))(d(Ψ) ∧ ρ) + β⁻¹ * Δ(ρ)

Lie

Source

Model

C::Form0
              
V::Form1
              
dX::Form1
              
V == ((⋆) ∘ (⋆))(C ∧ dX)

Mohamed Eq. 10, N2

Source

Model

(𝐮, w)::DualForm1
              
(P, 𝑝ᵈ)::DualForm0
              
μ::Constant
              
𝑝ᵈ == P + 0.5 * ι₁₁(w, w)
              
∂ₜ(𝐮) == μ * (∘(d, ⋆, d, ⋆))(w) + -1 * (⋆₁⁻¹)(w ∧ᵈᵖ₁₀ (⋆)(d(w))) + d(𝑝ᵈ)

Momentum

Source

Model

(f, b)::Form0
              
(v, V, g, Fᵥ, uˢ, v_up)::Form1
              
τ::Form2
              
U::Parameter
              
uˢ̇ == ∂ₜ(uˢ)
              
v_up == (((((((-1 * L(v, v) - L(V, v)) - L(v, V)) - f ∧ v) - (∘(⋆, d, ⋆))(uˢ) ∧ v) - d(p)) + b ∧ g) - (∘(⋆, d, ⋆))(τ)) + uˢ̇ + Fᵥ
              
uˢ̇ == force(U)

Navier-Stokes

Source

Partial differential equations which describe the motion of viscous fluid surfaces.

Model

(V, V̇, G)::Form1{X}
              
(ρ, ṗ, p)::Form0{X}
              
V̇ == neg₁(L₁′(V, V)) + div₁(kᵥ(Δ₁(V) + third(d₀(δ₁(V)))), avg₀₁(ρ)) + d₀(half(i₁′(V, V))) + neg₁(div₁(d₀(p), avg₀₁(ρ))) + G
              
∂ₜ(V) == V̇
              
ṗ == neg₀((⋆₀⁻¹)(L₀(V, (⋆₀)(p))))
              
∂ₜ(p) == ṗ

Oscillator

Source

Equation governing the motion of an object whose acceleration is negatively-proportional to its position.

Model

X::Form0
              
V::Form0
              
k::Constant
              
∂ₜ(X) == V
              
∂ₜ(V) == -k * X

Poiseuille

Source

A relation between the pressure drop in an incompressible and Newtownian fluid in laminar flow flowing through a long cylindrical pipe.

Model

P::Form0
              
q::Form1
              
(R, μ̃)::Constant
              
Δq == Δ(q)
              
∇P == d(P)
              
∂ₜ(q) == q̇
              
q̇ == μ̃ * ∂q(Δq) + ∇P + R * q

Poiseuille Density

Source

Model

q::Form1
              
(P, ρ)::Form0
              
(k, R, μ̃)::Constant
              
∂ₜ(q) == q̇
              
∇P == d(P)
              
q̇ == (μ̃ * ∂q(Δ(q)) - ∇P) + R * q
              
P == k * ρ
              
∂ₜ(ρ) == ρ̇
              
ρ_up == (∘(⋆, d, ⋆))(-1 * (ρ ∧₀₁ q))
              
ρ̇ == ∂ρ(ρ_up)

Schroedinger

Source

The evolution of the wave function over time.

Model

(i, h, m)::Constant
              
V::Parameter
              
Ψ::Form0
              
∂ₜ(Ψ) == (((-1 * h ^ 2) / (2m)) * Δ(Ψ) + V * Ψ) / (i * h)

Superposition

Source

Model

(C, Ċ)::Form0
              
(ϕ, ϕ₁, ϕ₂)::Form1
              
ϕ == ϕ₁ + ϕ₂
              
Ċ == (⋆₀⁻¹)(dual_d₁((⋆₁)(ϕ)))
              
∂ₜ(C) == Ċ

function findunreachabletvars(d)

Determine if the given Decapode can be compiled to an explicit time-stepping simulation. Use the simple check that one can traverse the Decapode starting from the state variables and reach all TVars (ignoring ∂ₜ edges).

function gensim(d::AbstractNamedDecapode; dimension::Int=2)

Generate a simulation function from the given Decapode. The returned function can then be combined with a mesh and a function describing function mappings to return a simulator to be passed to solve.

Gray-Scott

Source

A model of reaction-diffusion

Model

(U, V)::Form0
              
UV2::Form0
              
(U̇, V̇)::Form0
              
(f, k, rᵤ, rᵥ)::Constant
              
UV2 == U .* (V .* V)
              
U̇ == (rᵤ * Δ(U) - UV2) + f * (1 .- U)
              
V̇ == (rᵥ * Δ(V) + UV2) - (f + k) .* V
              
∂ₜ(U) == U̇
              
∂ₜ(V) == V̇

Brusselator

Source

A model of reaction-diffusion for an oscillatory chemical system.

Model

(U, V)::Form0
              
U2V::Form0
              
(U̇, V̇)::Form0
              
α::Constant
              
F::Parameter
              
U2V == (U .* U) .* V
              
U̇ == ((1 + U2V) - 4.4U) + α * Δ(U) + F
              
V̇ == (3.4U - U2V) + α * Δ(V)
              
∂ₜ(U) == U̇
              
∂ₜ(V) == V̇