7.1. Mathematical descriptions of boundary conditions

There are many types of boundary conditions. The most common boundary conditions for geophysical fluid dynamics are Dirichlet, von Neumann, and Robin boundary conditions.

7.1.1. Dirichlet boundary conditions

Dirichlet boundary conditions describes a boundary where you set a fixed value at the boundary. If you, for example, think about a 1D model for momentum \(u(x,t)\) defined in the range \(x\in[0,L]\) we can express this as:

(7.1)\[\begin{split}\begin{aligned} u(x_{0},t)=f_1(t),\\ u(x_{L},t)=f_2(t), \end{aligned}\end{split}\]

where \(x_{0,L}\) denotes the boundaries at \(x=0\) and \(x=L\), and \(f_{1,2}(t)\) is a function of time or a constant value.

Note

If \(f_{1,2}(t)\equiv 0\), we call it a homogenous boundary condition.

7.1.2. von Neuman boundary conditions

von Neumann boundary conditions describe a boundary where you define a fixed gradient, \(\frac{\partial u}{\partial x}\), at the boundary. For the same type of probem as above, this will be expressed as:

(7.2)\[\begin{split}\begin{aligned} \frac{\partial u(x_{0},t)}{\partial x}=f_1(t),\\ \frac{\partial u(x_{L},t)}{\partial x}=f_2(t) \end{aligned}\end{split}\]

7.1.3. Robin boundary conditions

Robin boundary conditions is a mix of von Neuman and Dirichlet boundary conditions, where you decide a linear combination of \(u(x,t)\) and the gradient \(\frac{\partial u}{\partial x}\) at the boundaries:

(7.3)\[\begin{split}\begin{aligned} \frac{\partial u(x_{0},t)}{\partial x}+a u(x_{0},t)=f1(t),\\ \frac{\partial u(x_{L},t)}{\partial x}+u(x_{L},t)=f_2(t) \end{aligned}\end{split}\]

You can read more about this in, e.g., Strauss [Str92].