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# pdepe

Solve initial-boundary value problems for parabolic-elliptic PDEs in 1-D

## Syntax

`sol = pdepe(m,pdefun,icfun,bcfun,xmesh,tspan)sol = pdepe(m,pdefun,icfun,bcfun,xmesh,tspan,options)[sol,tsol,sole,te,ie] = pdepe(m,pdefun,icfun,bcfun,xmesh,tspan,options)`

## Arguments

 `m` A parameter corresponding to the symmetry of the problem. `m` can be slab = `0`, cylindrical = `1`, or spherical = `2`. `pdefun` A handle to a function that defines the components of the PDE. `icfun` A handle to a function that defines the initial conditions. `bcfun` A handle to a function that defines the boundary conditions. `xmesh` A vector [`x0`, `x1`, ..., `xn`] specifying the points at which a numerical solution is requested for every value in `tspan`. The elements of `xmesh` must satisfy `x0 < x1 < ... < xn`. The length of `xmesh` must be >`= 3`. `tspan` A vector [`t0`, `t1`, ..., `tf`] specifying the points at which a solution is requested for every value in `xmesh`. The elements of `tspan` must satisfy `t0 < t1 < ... < tf`. The length of `tspan` must be `>=` `3`. `options` Some options of the underlying ODE solver are available in `pdepe`: `RelTol`, `AbsTol`, `NormControl`, `InitialStep`, `MaxStep`, and `Events`. In most cases, default values for these options provide satisfactory solutions. See `odeset` for details.

## Description

`sol = pdepe(m,pdefun,icfun,bcfun,xmesh,tspan)` solves initial-boundary value problems for systems of parabolic and elliptic PDEs in the one space variable x and time t. `pdefun`, `icfun`, and `bcfun` are function handles. See Create Function Handle for more information. The ordinary differential equations (ODEs) resulting from discretization in space are integrated to obtain approximate solutions at times specified in `tspan`. The `pdepe` function returns values of the solution on a mesh provided in `xmesh`.

Parameterizing Functions explains how to provide additional parameters to the functions `pdefun`, `icfun`, or `bcfun`, if necessary.

`pdepe` solves PDEs of the form:

 $c\left(x,t,u,\frac{\partial u}{\partial x}\right)\frac{\partial u}{\partial t}={x}^{-m}\frac{\partial }{\partial x}\left({x}^{m}f\left(x,t,u,\frac{\partial u}{\partial x}\right)\right)+s\left(x,t,u,\frac{\partial u}{\partial x}\right)$ (1-3)

The PDEs hold for t0ttf and axb. The interval [a,b] must be finite. m can be 0, 1, or 2, corresponding to slab, cylindrical, or spherical symmetry, respectively. If m > 0, then a must be ≥ 0.

In Equation 1-3, f (x,t,u,∂u/∂x) is a flux term and s (x,t,u,∂u/∂x) is a source term. The coupling of the partial derivatives with respect to time is restricted to multiplication by a diagonal matrix c (x,t,u,∂u/∂x). The diagonal elements of this matrix are either identically zero or positive. An element that is identically zero corresponds to an elliptic equation and otherwise to a parabolic equation. There must be at least one parabolic equation. An element of c that corresponds to a parabolic equation can vanish at isolated values of x if those values of x are mesh points. Discontinuities in c and/or s due to material interfaces are permitted provided that a mesh point is placed at each interface.

For t = t0 and all x, the solution components satisfy initial conditions of the form

 $u\left(x,{t}_{0}\right)={u}_{0}\left(x\right)$ (1-4)

For all t and either x = a or x = b, the solution components satisfy a boundary condition of the form

 $p\left(x,t,u\right)+q\left(x,t\right)f\left(x,t,u,\frac{\partial u}{\partial x}\right)=0$ (1-5)

Elements of q are either identically zero or never zero. Note that the boundary conditions are expressed in terms of the flux f rather than ∂u/∂x. Also, of the two coefficients, only p can depend on u.

In the call `sol = pdepe(m,pdefun,icfun,bcfun,xmesh,tspan)`:

• `m` corresponds to m.

• `xmesh(1)` and `xmesh(end)` correspond to a and b.

• `tspan(1)` and `tspan(end)` correspond to t0 and tf.

• `pdefun` computes the terms c, f, and s (Equation 1-3). It has the form

```[c,f,s] = pdefun(x,t,u,dudx) ```

The input arguments are scalars `x` and `t` and vectors `u` and `dudx` that approximate the solution u and its partial derivative with respect to x, respectively. `c`, `f`, and `s` are column vectors. `c` stores the diagonal elements of the matrix c (Equation 1-3).

• `icfun` evaluates the initial conditions. It has the form

```u = icfun(x) ```

When called with an argument `x`, `icfun` evaluates and returns the initial values of the solution components at `x` in the column vector `u`.

• `bcfun` evaluates the terms p and q of the boundary conditions (Equation 1-5). It has the form

```[pl,ql,pr,qr] = bcfun(xl,ul,xr,ur,t) ```

`ul` is the approximate solution at the left boundary `xl` = a and `ur` is the approximate solution at the right boundary `xr` = b. `pl` and `ql` are column vectors corresponding to p and q evaluated at `xl`, similarly `pr` and `qr` correspond to `xr`. When m > 0 and a = 0, boundedness of the solution near x = 0 requires that the flux f vanish at a = 0. `pdepe` imposes this boundary condition automatically and it ignores values returned in `pl` and `ql`.

`pdepe` returns the solution as a multidimensional array `sol`. ui = `ui` = `sol`(`:`,`:`,`i`) is an approximation to the `i`th component of the solution vector u. The element `ui`(`j`,`k`) = `sol`(`j`,`k`,`i`) approximates ui at (t,x) = (`tspan`(`j`),`xmesh`(`k`)).

`ui` = `sol`(`j`,`:`,`i`) approximates component `i` of the solution at time `tspan`(`j`) and mesh points `xmesh(:)`. Use `pdeval` to compute the approximation and its partial derivative ∂ui/∂x at points not included in `xmesh`. See `pdeval` for details.

`sol = pdepe(m,pdefun,icfun,bcfun,xmesh,tspan,options)` solves as above with default integration parameters replaced by values in `options`, an argument created with the `odeset` function. Only some of the options of the underlying ODE solver are available in `pdepe`: `RelTol`, `AbsTol`, `NormControl`, `InitialStep`, and `MaxStep`. The defaults obtained by leaving off the input argument `options` will generally be satisfactory. See `odeset` for details.

`[sol,tsol,sole,te,ie] = pdepe(m,pdefun,icfun,bcfun,xmesh,tspan,options)` with the `'Events'` property in `options` set to a function handle `Events`, solves as above while also finding where event functions `g(t,u(x,t))`are zero. For each function you specify whether the integration is to terminate at a zero and whether the direction of the zero crossing matters. Three column vectors are returned by `events`: `[value,isterminal,direction] = events(m,t,xmesh,umesh)`. `xmesh` contains the spatial mesh and `umesh` is the solution at the mesh points. Use `pdeval` to evaluate the solution between mesh points. For the I-th event function, `value(i)` is the value of the function, `ISTERMINAL(I) = 1` if the integration is to terminate at a zero of this event function and 0 otherwise. `direction(i) = 0` if all zeros are to be computed (the default), +1 if only zeros where the event function is increasing, and -1 if only zeros where the event function is decreasing. Output `tsol` is a column vector of times specified in `tspan`, prior to first terminal event. `SOL(j,:,:)` is the solution at `T(j)`. `TE` is a vector of times at which events occur. `SOLE(j,:,:)` is the solution at `TE(j)` and indices in vector `IE` specify which event occurred.

If `UI = SOL(j,:,i)` approximates component i of the solution at time `TSPAN(j)` and mesh points `XMESH`, `pdeval` evaluates the approximation and its partial derivative ∂ui/∂x at the array of points `XOUT` and returns them in `UOUT` and `DUOUTDX: [UOUT,DUOUTDX] = PDEVAL(M,XMESH,UI,XOUT)`

 Note:   The partial derivative ∂ui/∂x is evaluated here rather than the flux. The flux is continuous, but at a material interface the partial derivative may have a jump.

## Examples

Example 1. This example illustrates the straightforward formulation, computation, and plotting of the solution of a single PDE.

`${\pi }^{2}\frac{\partial u}{\partial t}=\frac{\partial }{\partial x}\left(\frac{\partial u}{\partial x}\right)$`

This equation holds on an interval 0 ≤ x ≤ 1 for times t ≥ 0.

The PDE satisfies the initial condition

`$u\left(x,0\right)=\mathrm{sin}\pi x$`

and boundary conditions

`$\begin{array}{l}u\left(0,t\right)\equiv 0\hfill \\ \pi {e}^{-t}+\frac{\partial u}{\partial x}\left(1,t\right)=0\hfill \end{array}$`

It is convenient to use local functions to place all the functions required by `pdepe` in a single function.

```function pdex11 m = 0; x = linspace(0,1,20); t = linspace(0,2,5); sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t); % Extract the first solution component as u. u = sol(:,:,1); % A surface plot is often a good way to study a solution. surf(x,t,u) title('Numerical solution computed with 20 mesh points.') xlabel('Distance x') ylabel('Time t') % A solution profile can also be illuminating. figure plot(x,u(end,:)) title('Solution at t = 2') xlabel('Distance x') ylabel('u(x,2)') % -------------------------------------------------------------- function [c,f,s] = pdex1pde(x,t,u,DuDx) c = pi^2; f = DuDx; s = 0; % -------------------------------------------------------------- function u0 = pdex1ic(x) u0 = sin(pi*x); % -------------------------------------------------------------- function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t) pl = ul; ql = 0; pr = pi * exp(-t); qr = 1; ```

In this example, the PDE, initial condition, and boundary conditions are coded in local functions `pdex1pde`, `pdex1ic`, and `pdex1bc`.

The surface plot shows the behavior of the solution.

The following plot shows the solution profile at the final value of `t` (i.e., `t = 2`).

Example 2. This example illustrates the solution of a system of PDEs. The problem has boundary layers at both ends of the interval. The solution changes rapidly for small t.

The PDEs are

`$\begin{array}{c}\frac{\partial {u}_{1}}{\partial t}=0.024\frac{{\partial }^{2}{u}_{1}}{\partial {x}^{2}}-F\left({u}_{1}-{u}_{2}\right)\\ \frac{\partial {u}_{2}}{\partial t}=0.170\frac{{\partial }^{2}{u}_{2}}{\partial {x}^{2}}+F\left({u}_{1}-{u}_{2}\right)\end{array}$`

where F(y) = exp(5.73y) – exp(–11.46y).

This equation holds on an interval 0 ≤ x ≤ 1 for times t ≥ 0.

The PDE satisfies the initial conditions

`$\begin{array}{c}{u}_{1}\left(x,0\right)\equiv 1\\ {u}_{2}\left(x,0\right)\equiv 0\end{array}$`

and boundary conditions

`$\begin{array}{l}\frac{\partial {u}_{1}}{\partial x}\left(0,t\right)\equiv 0\hfill \\ {u}_{2}\left(0,t\right)\equiv 0\hfill \\ {u}_{1}\left(1,t\right)\equiv 1\hfill \\ \frac{\partial {u}_{2}}{\partial x}\left(1,t\right)\equiv 0\hfill \end{array}$`

In the form expected by `pdepe`, the equations are

`$\left[\begin{array}{l}1\\ 1\end{array}\right]\cdot *\frac{\partial }{\partial t}\left[\begin{array}{l}{u}_{1}\\ {u}_{2}\end{array}\right]=\frac{\partial }{\partial x}\left[\begin{array}{l}0.024\left(\partial {u}_{1}\text{/}\partial x\right)\\ 0.170\left(\partial {u}_{2}\text{/}\partial x\right)\end{array}\right]+\left[\begin{array}{r}-F\left({u}_{1}-{u}_{2}\right)\\ F\left({u}_{1}-{u}_{2}\right)\end{array}\right]$`

The boundary conditions on the partial derivatives of u have to be written in terms of the flux. In the form expected by `pdepe`, the left boundary condition is

`$\left[\begin{array}{c}0\\ {u}_{2}\end{array}\right]+\left[\begin{array}{l}1\\ 0\end{array}\right]\cdot *\left[\begin{array}{l}0.024\left(\partial {u}_{1}\text{/}\partial x\right)\\ 0.170\left(\partial {u}_{2}\text{/}\partial x\right)\end{array}\right]=\left[\begin{array}{l}0\\ 0\end{array}\right]$`

and the right boundary condition is

`$\left[\begin{array}{c}{u}_{1}-1\\ 0\end{array}\right]+\left[\begin{array}{l}0\\ 1\end{array}\right]\cdot *\left[\begin{array}{l}0.024\left(\partial {u}_{1}\text{/}\partial x\right)\\ 0.170\left(\partial {u}_{2}\text{/}\partial x\right)\end{array}\right]=\left[\begin{array}{l}0\\ 0\end{array}\right]$`

The solution changes rapidly for small t. The program selects the step size in time to resolve this sharp change, but to see this behavior in the plots, the example must select the output times accordingly. There are boundary layers in the solution at both ends of [0,1], so the example places mesh points near `0` and `1` to resolve these sharp changes. Often some experimentation is needed to select a mesh that reveals the behavior of the solution.

```function pdex4 m = 0; x = [0 0.005 0.01 0.05 0.1 0.2 0.5 0.7 0.9 0.95 0.99 0.995 1]; t = [0 0.005 0.01 0.05 0.1 0.5 1 1.5 2]; sol = pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t); u1 = sol(:,:,1); u2 = sol(:,:,2); figure surf(x,t,u1) title('u1(x,t)') xlabel('Distance x') ylabel('Time t') figure surf(x,t,u2) title('u2(x,t)') xlabel('Distance x') ylabel('Time t') % -------------------------------------------------------------- function [c,f,s] = pdex4pde(x,t,u,DuDx) c = [1; 1]; f = [0.024; 0.17] .* DuDx; y = u(1) - u(2); F = exp(5.73*y)-exp(-11.47*y); s = [-F; F]; % -------------------------------------------------------------- function u0 = pdex4ic(x); u0 = [1; 0]; % -------------------------------------------------------------- function [pl,ql,pr,qr] = pdex4bc(xl,ul,xr,ur,t) pl = [0; ul(2)]; ql = [1; 0]; pr = [ur(1)-1; 0]; qr = [0; 1]; ```

In this example, the PDEs, initial conditions, and boundary conditions are coded in local functions `pdex4pde`, `pdex4ic`, and `pdex4bc`.

The surface plots show the behavior of the solution components.

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### Tips

• The arrays `xmesh` and `tspan` play different roles in `pdepe`.

`tspan` – The `pdepe` function performs the time integration with an ODE solver that selects both the time step and formula dynamically. The elements of `tspan` merely specify where you want answers and the cost depends weakly on the length of `tspan`.

`xmesh` – Second order approximations to the solution are made on the mesh specified in `xmesh`. Generally, it is best to use closely spaced mesh points where the solution changes rapidly. `pdepe` does not select the mesh in x automatically. You must provide an appropriate fixed mesh in `xmesh`. The cost depends strongly on the length of `xmesh`. When m > 0, it is not necessary to use a fine mesh near x = 0 to account for the coordinate singularity.

• The time integration is done with `ode15s`. `pdepe` exploits the capabilities of `ode15s` for solving the differential-algebraic equations that arise when Equation 1-3 contains elliptic equations, and for handling Jacobians with a specified sparsity pattern.

• After discretization, elliptic equations give rise to algebraic equations. If the elements of the initial conditions vector that correspond to elliptic equations are not "consistent" with the discretization, `pdepe` tries to adjust them before beginning the time integration. For this reason, the solution returned for the initial time may have a discretization error comparable to that at any other time. If the mesh is sufficiently fine, `pdepe` can find consistent initial conditions close to the given ones. If `pdepe` displays a message that it has difficulty finding consistent initial conditions, try refining the mesh.

No adjustment is necessary for elements of the initial conditions vector that correspond to parabolic equations.

## References

[1] Skeel, R. D. and M. Berzins, "A Method for the Spatial Discretization of Parabolic Equations in One Space Variable," SIAM Journal on Scientific and Statistical Computing, Vol. 11, 1990, pp.1–32.