1D Heat Equation

1D Heat Equation#

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

\[\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}\]

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

Domain and Boundary Conditions#

The domain is \(x \in [0, 1]\) with Dirichlet boundary conditions:

  • \(T(0, t) = 100\)

  • \(T(1, t) = 100\)

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 time step is constrained by the stability condition for the explicit scheme: $\(\Delta t \leq \frac{\Delta x^2}{3\alpha}\)$

This example is implemented in:

Results#

The solution shows the heat diffusing through the domain, with the temperature at the boundaries held constant at 100. The explicit scheme is conditionally stable, requiring a small time step, while the implicit scheme is unconditionally stable but requires solving a linear system at each time step.

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