2D Heat Equation

2D Heat Equation#

This example solves the two-dimensional heat equation with Dirichlet boundary conditions, which is a classic parabolic PDE:

\[ \frac{\partial T}{\partial t} = \alpha\left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} \right) \]

where \(T\) is the temperature and \(\alpha\) is the thermal diffusivity.

Domain and Boundary Conditions#

The domain is \((x,y) \in [0,2]\times[0,2]\) with Dirichlet boundary conditions:

  • \(T(0, y, t) = 0\)

  • \(T(2, y, t) = 0\)

  • \(T(x, 0, t) = 0\)

  • \(T(x, 2, t) = 0\)

The initial condition is a square region with value \(T=2\), while the rest of the domain is initialized to \(T=0\).

Discretization#

The spatial discretization uses the mimetic Laplacian operator with a specified order of accuracy \(k\). The temporal discretization can be either:

  1. Explicit, forward Euler: \(T^{n+1} = T^n + \alpha \Delta t L T^n\)

  2. Implicit, backward Euler: \(T^{n+1} = (I - \alpha \Delta t L)^{-1} T^n\)

where \(L\) is the mimetic discrete Laplacian operator.

The explicit scheme requires a sufficiently small time step for stability.

This example is implemented in:

Results#

The solution shows the initial pulse diffusing through the two-dimensional domain, with the temperature at the boundaries held constant at zero. The explicit scheme is conditionally stable.

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