Burgers' equation: Difference between revisions
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=The equation= | |||
Let us consider the Burger's equation, which describes the dynamics of viscous fluid without the effects of pressure. Despite being unrealistic it is the simplest description of advective flow with diffusive effects of viscosity. It has the following form | |||
\begin{equation} | |||
\frac{\partial \b{u}}{\partial t} + (\b{u}\cdot\nabla)\b{u} = \nu \nabla^2 \b{u} | |||
\end{equation} | |||
where $\b{u}$ is a velocity field. For time stepping the advection term must be linearized. There are several tactics of linearization, but we choose the simplest one, which is using the current velocity as an approximant of the velociy in the next time step. Using the euler step method and the implicit solving scheme we arive at the equation | |||
\begin{equation} | |||
\b{u}^{n+1}_i + [\b{u}^{n}_i\cdot\nabla_i\b{u}^{n+1} - \nu\nabla^2_i \b{u}^{n+1}]\Delta t = \b{u}^{n}_i | |||
\end{equation} | |||
$$\ | where the subscript $i$ is the index of the domain node, superscripts $n$ and $n+1$ denote the current and the next time step respectively, $\Delta t$ is the lenght of the time step. The notation $\nabla_i$ means the gradient operator at node $i$, not the component of the gradient. Same holds for $\nabla^2_i$. | ||
This equation can be solved for $\b{u}^{n+1}_i$ using the Medusa library. Let us look at an example. | |||
Revision as of 13:24, 12 January 2024
The equation
Let us consider the Burger's equation, which describes the dynamics of viscous fluid without the effects of pressure. Despite being unrealistic it is the simplest description of advective flow with diffusive effects of viscosity. It has the following form \begin{equation} \frac{\partial \b{u}}{\partial t} + (\b{u}\cdot\nabla)\b{u} = \nu \nabla^2 \b{u} \end{equation}
where $\b{u}$ is a velocity field. For time stepping the advection term must be linearized. There are several tactics of linearization, but we choose the simplest one, which is using the current velocity as an approximant of the velociy in the next time step. Using the euler step method and the implicit solving scheme we arive at the equation
\begin{equation} \b{u}^{n+1}_i + [\b{u}^{n}_i\cdot\nabla_i\b{u}^{n+1} - \nu\nabla^2_i \b{u}^{n+1}]\Delta t = \b{u}^{n}_i \end{equation}
where the subscript $i$ is the index of the domain node, superscripts $n$ and $n+1$ denote the current and the next time step respectively, $\Delta t$ is the lenght of the time step. The notation $\nabla_i$ means the gradient operator at node $i$, not the component of the gradient. Same holds for $\nabla^2_i$.
This equation can be solved for $\b{u}^{n+1}_i$ using the Medusa library. Let us look at an example.