Boundary Conditions

What follows is a set of allowable boundary conditions, please note that this is not exhaustive - try your condition and see if it works, the handling is quite general. If it doesn't please post an issue and we'll try to support it. At the moment boundary conditions have to be supplied at the edge of the domain, but there are plans to support conditions embedded in the domain.

Definitions

using ModelingToolkit, MethodOfLines, Domainsets

@parameters x y t
@variables u(..) v(..)
Dt = Differential(t)
Dx = Differential(x)
Dy = Differential(y)
Dxx = Differential(x)^2
Dyy = Differential(y)^2

x_min = y_min = 0.0

x_max = y_max = 1.0

Dirichlet

v(t, 0, y) ~ 1.0

Time dependant

u(t, 0., y) ~ x_min*y+ 0.5t

Julia function

v(t, x, y_max) ~ sin(x)

User defined function

alpha = 9

f(t,x,y) = x*y - t

function g(x,y) 
    z = sin(x*y)+cos(y)
    # Note that symbolic conditionals require the use of IfElse.ifelse, or registration
    return IfElse.ifelse(z > 0, x, 1.0)
end

u(t,x,y_min) ~ f(t,x,y_min) + alpha/g(x,y_min)

Registered User Defined Function

alpha = 9

f(t,x,y) = x*y - t

function g(x,y) 
    z = sin(x*y)+cos(y)
    # This function must be registered as it contains a symbolic conditional
    if z > 0
        return x
    else
        return 1.0
    end
end

@register g(x, y)

u(t,x,y_min) ~ f(t,x,y_min) + alpha/g(x,y_min)

Neumann/Robin

v(t, x_min, y) ~ 2. * Dx(v(t, x_min, y))

Time dependant

u(t, x_min, y) ~ x_min*Dy(v(t,x_min,y)) + 0.5t

Higher order

v(t, x, 1.0) ~ sin(x) + Dyy(v(t, x, y_max))

Time derivative

Dt(u(t, x_min, y)) ~ 0.2

User defined function

function f(u, v)
    (u + Dyy(v) - Dy(u))/(1 + v)
end

Dyy(u(t, x, y_min)) ~ f(u(t, x, y_min), v(t, x, y_min)) + 1

0 lhs

0 ~ u(t, x, y_max) - Dy(v(t, x, y_max))

Periodic

u(t, x_min, y) ~ u(t, x_max, y)

v(t, x, y_max) ~ u(t, x_max, y)