Quick start

    VisualPDE is a web-based set of tools for solving partial differential equations (PDEs) via an interactive, easy-to-use simulation. To get started, try playing with some of the linear examples, or read on for some quick tips for using the solver.

    Interacting with the simulation

    Clicking/pressing on the simulation draws values right onto the domain. You can customise exactly what this does under → Brush For example, the default settings in the heat equation example allow you to paint ‘heat’ of value 1 onto the domain, which acts like an initial condition for the rest of the simulation.

    The equations panel

    Pressing   opens up the equations panel.

    Equations panel

    Here you can:

    Domain shape

    The default domain for solving PDEs is a 2D rectangle, $\domain = [0,L_x]\times[0,L_y]$, which fits the size of your browser window or phone screen. Throughout VisualPDE, we use coordinates $x\in[0,L_x]$ and $y\in[0,L_y]$.

    You can force the domain to be a square, $\domain = [0,L]\times[0,L]$, by toggling off → Domain → Fill screen

    Boundary conditions

    The following boundary conditions are available to allow you to set the value of the function, or the value of its derivative, along the boundary $\boundary$ of the domain $\domain$:

    You can swap between boundary conditions by choosing   → Boundary conditions and selecting from the list for each variable.

    Initial conditions

    You can specify the values to which the unknowns ($u$, $v$, $w$) are initialised when resetting the simulation. These expressions can be functions of $x$, $y$, the special string ‘RAND’ that assigns a random number in [0,1] to each point in the domain, along with any user-defined parameters and the images $I_S$ and $I_T$ (see the advanced documentation for more details). You can also use $L$, $L_x$ and $L_y$.

    Changing the equations

    The simplest system VisualPDE can solve is a single PDE,

    \[\pd{u}{t} = \vnabla \cdot (D_u \vnabla u) + f_u,\]

    where $D_u$ and $f_u$ are functions of $u$, $x$, $y$, and $t$ that you can specify.

    The most complicated type is a coupled system of PDEs in four unknowns, $u$, $v$, $w$ and $q$:

    \[\begin{aligned} t_u\pd{u}{t} &= \vnabla \cdot(D_{uu}\vnabla u+D_{uv}\vnabla v+D_{uw}\vnabla w+D_{uq}\vnabla q) + f_u,\\ \text{one of}\left\{\begin{matrix}\displaystyle t_v\pd{v}{t} \\ v\end{matrix}\right. & \begin{aligned} &= \vnabla \cdot(D_{vu}\vnabla u+D_{vv}\vnabla v+D_{vw}\vnabla w+D_{vq}\vnabla q) + f_v \vphantom{\displaystyle t_v\pd{v}{t}}, \\ &= \vnabla \cdot(D_{vu}\vnabla u+D_{vw}\vnabla w+D_{vq}\vnabla q) + f_v, \end{aligned}\\ \text{one of}\left\{\begin{matrix}\displaystyle t_w\pd{w}{t} \\ w\end{matrix}\right. & \begin{aligned} &= \vnabla \cdot(D_{wu}\vnabla u+D_{wv}\vnabla v+D_{ww}\vnabla w+D_{wq}\vnabla q) + f_w \vphantom{\displaystyle t_w\pd{w}{t}}, \\ &= \vnabla \cdot(D_{wu}\vnabla u+D_{wv}\vnabla v+D_{wq}\vnabla q) + f_w, \end{aligned}\\ \text{one of}\left\{\begin{matrix}\displaystyle t_q\pd{q}{t} \\ q\end{matrix}\right. & \begin{aligned} &= \vnabla \cdot(D_{qu}\vnabla u+D_{qv}\vnabla v+D_{qw}\vnabla w+D_{qq}\vnabla q) + f_q \vphantom{\displaystyle t_q\pd{q}{t}}, \\ &= \vnabla \cdot(D_{qu}\vnabla u+D_{qv}\vnabla v+D_{qw}\vnabla w) + f_q, \end{aligned} \end{aligned}\]

    where $D_{uu}, \dots, D_{qq}$, $f_u, \dots, f_q$ and $t_u, \dots, t_q$ are functions of $u$, $v$, $w$, $q$, $x$, $y$ and $t$ that you can specify.

    More VisualPDE

    For a comprehensive list of all the options that you can set in VisualPDE, check out the Advanced documentation, or discover what VisualPDE can solve in our brief summary.