Upwind finite difference schemes

struct HamiltonJacobiEvolution{O}

A standard forward Euler in time finite difference method for solving the Hamilton-Jacobi evolution equation and reinitialisation equation on order O finite elements in serial or parallel.

Based on the scheme by Osher and Fedkiw (link).

Parameters

• stencil::Stencil: Spatial finite difference stencil for a single step HJ equation and reinitialisation equation.
• model: A CartesianDiscreteModel.
• space: FE space for level-set function
• perm: A permutation vector
• params: Tuple of additional params
• cache: Stencil cache

HamiltonJacobiEvolution(stencil::Stencil,model,space,tol,max_steps,max_steps_reinit)

Create an instance of HamiltonJacobiEvolution given a stencil, model, FE space, and additional optional arguments. This automatically creates the DoF permutation to handle high-order finite elements.

evolve!(::LevelSetEvolution,φ,args...)

Evolve the level set function φ according to an evolution method LevelSetEvolution.

evolve!(::Stencil,φ,vel,Δt,Δx,isperiodic,caches) -> φ

Single finite difference step of the Hamilton-Jacobi evoluation equation for a given Stencil.

evolve!(s::HamiltonJacobiEvolution{O},φ,vel,γ) where O

Solve the Hamilton-Jacobi evolution equation using the HamiltonJacobiEvolution.

Hamilton-Jacobi evolution equation

$\frac{\partial\phi}{\partial t} + V(\boldsymbol{x})\lVert\boldsymbol{\nabla}\phi\rVert = 0,$

with $\phi(0,\boldsymbol{x})=\phi_0(\boldsymbol{x})$ and $\boldsymbol{x}\in D,~t\in(0,T)$.

Arguments

• s::HamiltonJacobiEvolution{O}: Method
• φ: level set function as a vector of degrees of freedom
• vel: the normal velocity as a vector of degrees of freedom
• γ: coeffient on the time step size.

reinit!(::LevelSetEvolution,φ,args...)

Reinitialise the level set function φ according to an evolution method LevelSetEvolution.

reinit!(::Stencil,φ_new,φ,vel,Δt,Δx,isperiodic,caches) -> φ

Single finite difference step of the reinitialisation equation for a given Stencil.

reinit!(s::HamiltonJacobiEvolution{O},φ,γ) where O

Solve the reinitialisation equation using the HamiltonJacobiEvolution.

Reinitialisation equation

$\frac{\partial\phi}{\partial t} + \mathrm{sign}(\phi_0)(\lVert\boldsymbol{\nabla}\phi\rVert-1) = 0,$

with $\phi(0,\boldsymbol{x})=\phi_0(\boldsymbol{x})$ and $\boldsymbol{x}\in D,~t\in(0,T)$.

Arguments

• s::HamiltonJacobiEvolution{O}: Method
• φ: level set function as a vector of degrees of freedom
• γ: coeffient on the time step size.

get_dof_Δ(m::HamiltonJacobiEvolution)

Return the distance betweem degree of freedom

struct FirstOrderStencil{D,T} <: Stencil end

A first order upwind difference scheme based on Osher and Fedkiw (link).

abstract type Stencil

Finite difference stencil for a single step of the Hamilton-Jacobi evolution equation and reinitialisation equation.

Your own spatial stencil can be implemented by extending the methods below.

evolve!(::Stencil,φ,vel,Δt,Δx,isperiodic,caches) -> φ

Single finite difference step of the Hamilton-Jacobi evoluation equation for a given Stencil.

reinit!(::Stencil,φ_new,φ,vel,Δt,Δx,isperiodic,caches) -> φ

Single finite difference step of the reinitialisation equation for a given Stencil.

allocate_caches(::Stencil,φ,vel)

Allocate caches for a given Stencil.

check_order(::Stencil,order)

Throw error if insufficient reference element order to implement stencil in parallel.