# parabolic

(Not recommended) Solve parabolic PDE problem

`parabolic`

is not recommended. Use `solvepde`

instead.

## Syntax

## Description

Parabolic equation solver

Solves PDE problems of the type

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\nabla u\right)+au=f$$

on a 2-D or 3-D region Ω, or the system PDE problem

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

The variables *c*, *a*, *f*, and
*d* can depend on position, time, and the solution *u* and
its gradient.

produces the solution to the FEM formulation of the scalar PDE problem`u`

= parabolic(`u0`

,`tlist`

,`model`

,`c`

,`a`

,`f`

,`d`

)

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\nabla u\right)+au=f$$

on a 2-D or 3-D region Ω, or the system PDE problem

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

with geometry, mesh, and boundary conditions specified in `model`

, and
with initial value `u0`

. The variables *c*,
*a*, *f*, and *d* in the equation
correspond to the function coefficients `c`

, `a`

,
`f`

, and `d`

respectively.

,
for any of the previous input arguments, turns off the display of internal ODE solver
statistics during the solution process.`u`

= parabolic(___,'Stats','off')

## Examples

### Parabolic Equation

Solve the parabolic equation

$$\frac{\partial u}{\partial t}=\Delta u$$

on the square domain specified by `squareg`

.

Create a PDE model and import the geometry.

model = createpde; geometryFromEdges(model,@squareg); pdegplot(model,'EdgeLabels','on') ylim([-1.1,1.1]) axis equal

Set Dirichlet boundary conditions $$u=0$$ on all edges.

applyBoundaryCondition(model,'dirichlet',... 'Edge',1:model.Geometry.NumEdges, ... 'u',0);

Generate a relatively fine mesh.

generateMesh(model,'Hmax',0.02,'GeometricOrder','linear');

Set the initial condition to have $$u(0)=1$$ on the disk $${x}^{2}+{y}^{2}\le 0.{4}^{2}$$ and $$u(0)=0$$ elsewhere.

p = model.Mesh.Nodes; u0 = zeros(size(p,2),1); ix = find(sqrt(p(1,:).^2 + p(2,:).^2) <= 0.4); u0(ix) = ones(size(ix));

Set solution times to be from 0 to 0.1 with step size 0.005.

tlist = linspace(0,0.1,21);

Create the PDE coefficients.

c = 1; a = 0; f = 0; d = 1;

Solve the PDE.

u = parabolic(u0,tlist,model,c,a,f,d);

134 successful steps 0 failed attempts 270 function evaluations 1 partial derivatives 26 LU decompositions 269 solutions of linear systems

Plot the initial condition, the solution at the final time, and two intermediate solutions.

figure subplot(2,2,1) pdeplot(model,'XYData',u(:,1)); axis equal title('t = 0') subplot(2,2,2) pdeplot(model,'XYData',u(:,5)) axis equal title('t = 0.02') subplot(2,2,3) pdeplot(model,'XYData',u(:,11)) axis equal title('t = 0.05') subplot(2,2,4) pdeplot(model,'XYData',u(:,end)) axis equal title('t = 0.1')

### Parabolic Equation Using Legacy Syntax

Solve the parabolic equation

$$\frac{\partial u}{\partial t}=\Delta u$$

on the square domain specified by `squareg`

, using a geometry function to specify the geometry, a boundary function to specify the boundary conditions, and using `initmesh`

to create the finite element mesh.

Specify the geometry as `@squareg`

and plot the geometry.

g = @squareg; pdegplot(g,'EdgeLabels','on') ylim([-1.1,1.1]) axis equal

Set Dirichlet boundary conditions $$u=0$$ on all edges. The `squareb1`

function specifies these boundary conditions.

b = @squareb1;

Generate a relatively fine mesh.

`[p,e,t] = initmesh(g,'Hmax',0.02);`

Set the initial condition to have $$u(0)=1$$ on the disk $${x}^{2}+{y}^{2}\le 0.{4}^{2}$$ and $$u(0)=0$$ elsewhere.

u0 = zeros(size(p,2),1); ix = find(sqrt(p(1,:).^2 + p(2,:).^2) <= 0.4); u0(ix) = ones(size(ix));

Set solution times to be from 0 to 0.1 with step size 0.005.

tlist = linspace(0,0.1,21);

Create the PDE coefficients.

c = 1; a = 0; f = 0; d = 1;

Solve the PDE.

u = parabolic(u0,tlist,b,p,e,t,c,a,f,d);

147 successful steps 0 failed attempts 296 function evaluations 1 partial derivatives 28 LU decompositions 295 solutions of linear systems

Plot the initial condition, the solution at the final time, and two intermediate solutions.

figure subplot(2,2,1) pdeplot(p,e,t,'XYData',u(:,1)); axis equal title('t = 0') subplot(2,2,2) pdeplot(p,e,t,'XYData',u(:,5)) axis equal title('t = 0.02') subplot(2,2,3) pdeplot(p,e,t,'XYData',u(:,11)) axis equal title('t = 0.05') subplot(2,2,4) pdeplot(p,e,t,'XYData',u(:,end)) axis equal title('t = 0.1')

### Parabolic Problem Using Matrix Coefficients

Create finite element matrices that encode a parabolic problem, and solve the problem.

The problem is the evolution of temperature in a conducting block. The block is a rectangular slab.

model = createpde(1); importGeometry(model,'Block.stl'); handl = pdegplot(model,'FaceLabels','on'); view(-42,24) handl(1).FaceAlpha = 0.5;

Faces 1, 4, and 6 of the slab are kept at 0 degrees. The other faces are insulated. Include the boundary condition on faces 1, 4, and 6. You do not need to include the boundary condition on the other faces because the default condition is insulated.

applyBoundaryCondition(model,'dirichlet','Face',[1,4,6],'u',0);

The initial temperature distribution in the block has the form

$${u}_{0}=1{0}^{-3}xyz.$$

generateMesh(model); p = model.Mesh.Nodes; x = p(1,:); y = p(2,:); z = p(3,:); u0 = x.*y.*z*1e-3;

The parabolic equation in toolbox syntax is

$$d\frac{\partial u}{\partial t}-\nabla \cdot (c\nabla u)+au=f.$$

Suppose the thermal conductivity of the block leads to a $$c$$ coefficient value of 1. The values of the other coefficients in this problem are $$d=1$$, $$a=0$$, and $$f=0$$.

d = 1; c = 1; a = 0; f = 0;

Create the finite element matrices that encode the problem.

[Kc,Fc,B,ud] = assempde(model,c,a,f); [~,M,~] = assema(model,0,d,f);

Solve the problem at time steps of 1 for times ranging from 0 to 40.

tlist = linspace(0,40,41); u = parabolic(u0,tlist,Kc,Fc,B,ud,M);

35 successful steps 0 failed attempts 72 function evaluations 1 partial derivatives 10 LU decompositions 71 solutions of linear systems

Plot the solution on the outside of the block at times 0, 10, 25, and 40. Ensure that the coloring is the same for all plots.

umin = min(min(u)); umax = max(max(u)); subplot(2,2,1) pdeplot3D(model,'ColorMapData',u(:,1)) colorbar off view(125,22) title 't = 0' clim([umin umax]); subplot(2,2,2) pdeplot3D(model,'ColorMapData',u(:,11)) colorbar off view(125,22) title 't = 10' clim([umin umax]); subplot(2,2,3) pdeplot3D(model,'ColorMapData',u(:,26)) colorbar off view(125,22) title 't = 25' clim([umin umax]); subplot(2,2,4) pdeplot3D(model,'ColorMapData',u(:,41)) colorbar off view(125,22) title 't = 40' clim([umin umax]);

## Input Arguments

`u0`

— Initial condition

vector | character vector | character array | string scalar | string vector

Initial condition, specified as a scalar, vector of nodal values, character vector, character
array, string scalar, or string vector. The initial condition is the value of the
solution `u`

at the initial time, specified as a column vector of
values at the nodes. The nodes are either `p`

in the
`[p,e,t]`

data structure, or are
`model.Mesh.Nodes`

.

If the initial condition is a constant scalar

`v`

, specify`u0`

as`v`

.If there are

`Np`

nodes in the mesh, and*N*equations in the system of PDEs, specify`u0`

as a column vector of`Np`

**N*elements, where the first`Np`

elements correspond to the first component of the solution`u`

, the second`Np`

elements correspond to the second component of the solution`u`

, etc.Give a text expression of a function, such as

`'x.^2 + 5*cos(x.*y)'`

. If you have a system of*N*> 1 equations, give a text array such aschar('x.^2 + 5*cos(x.*y)',... 'tanh(x.*y)./(1+z.^2)')

**Example: **`x.^2+5*cos(y.*x)`

**Data Types: **`double`

| `char`

| `string`

**Complex Number Support: **Yes

`tlist`

— Solution times

real vector

Solution times, specified as a real vector. The solver returns the solution to the PDE at the solution times.

**Example: **`0:0.2:4`

**Data Types: **`double`

`model`

— PDE model

`PDEModel`

object

PDE model, specified as a `PDEModel`

object.

**Example: **`model = createpde`

`c`

— PDE coefficient

scalar | matrix | character vector | character array | string scalar | string vector | coefficient function

PDE coefficient, specified as a scalar, matrix, character vector, character array,
string scalar, string vector, or coefficient function. `c`

represents
the *c* coefficient in the scalar PDE

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\nabla u\right)+au=f$$

or in the system of PDEs

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

**Example: **`'cosh(x+y.^2)'`

**Data Types: **`double`

| `char`

| `string`

| `function_handle`

**Complex Number Support: **Yes

`a`

— PDE coefficient

scalar | matrix | character vector | character array | string scalar | string vector | coefficient function

PDE coefficient, specified as a scalar, matrix, character vector, character array,
string scalar, string vector, or coefficient function. `a`

represents
the *a* coefficient in the scalar PDE

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\nabla u\right)+au=f$$

or in the system of PDEs

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

**Example: **`2*eye(3)`

**Data Types: **`double`

| `char`

| `string`

| `function_handle`

**Complex Number Support: **Yes

`f`

— PDE coefficient

scalar | matrix | character vector | character array | string scalar | string vector | coefficient function

PDE coefficient, specified as a scalar, matrix, character vector, character array,
string scalar, string vector, or coefficient function. `f`

represents
the *f* coefficient in the scalar PDE

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\nabla u\right)+au=f$$

or in the system of PDEs

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

**Example: **`char('sin(x)';'cos(y)';'tan(z)')`

**Data Types: **`double`

| `char`

| `string`

| `function_handle`

**Complex Number Support: **Yes

`d`

— PDE coefficient

scalar | matrix | character vector | character array | string scalar | string vector | coefficient function

PDE coefficient, specified as a scalar, matrix, character vector, character array,
string scalar, string vector, or coefficient function. `d`

represents
the *d* coefficient in the scalar PDE

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\nabla u\right)+au=f$$

or in the system of PDEs

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

**Example: **`2*eye(3)`

**Data Types: **`double`

| `char`

| `string`

| `function_handle`

**Complex Number Support: **Yes

`b`

— Boundary conditions

boundary matrix | boundary file

Boundary conditions, specified as a boundary matrix or boundary file. Pass a boundary file as a function handle or as a file name. A boundary matrix is generally an export from the PDE Modeler app.

**Example: **`b = 'circleb1'`

, `b = "circleb1"`

, or ```
b =
@circleb1
```

**Data Types: **`double`

| `char`

| `string`

| `function_handle`

`p`

— Mesh points

matrix

Mesh points, specified as a 2-by-`Np`

matrix of points, where
`Np`

is the number of points in the mesh. For a description of the
(`p`

,`e`

,`t`

) matrices, see Mesh Data as [p,e,t] Triples.

Typically, you use the `p`

, `e`

, and `t`

data exported from the PDE Modeler app, or generated by `initmesh`

or `refinemesh`

.

**Example: **`[p,e,t] = initmesh(gd)`

**Data Types: **`double`

`e`

— Mesh edges

matrix

Mesh edges, specified as a `7`

-by-`Ne`

matrix of edges,
where `Ne`

is the number of edges in the mesh. For a description of the
(`p`

,`e`

,`t`

) matrices, see Mesh Data as [p,e,t] Triples.

Typically, you use the `p`

, `e`

, and `t`

data exported from the PDE Modeler app, or generated by `initmesh`

or `refinemesh`

.

**Example: **`[p,e,t] = initmesh(gd)`

**Data Types: **`double`

`t`

— Mesh triangles

matrix

Mesh triangles, specified as a `4`

-by-`Nt`

matrix of
triangles, where `Nt`

is the number of triangles in the mesh. For a
description of the (`p`

,`e`

,`t`

)
matrices, see Mesh Data as [p,e,t] Triples.

Typically, you use the `p`

, `e`

, and `t`

data exported from the PDE Modeler app, or generated by `initmesh`

or `refinemesh`

.

**Example: **`[p,e,t] = initmesh(gd)`

**Data Types: **`double`

`Kc`

— Stiffness matrix

sparse matrix | full matrix

Stiffness matrix, specified as a sparse matrix or as a full
matrix. See Elliptic Equations. Typically, `Kc`

is
the output of `assempde`

.

`Fc`

— Load vector

vector

Load vector, specified as a vector. See Elliptic Equations. Typically, `Fc`

is the
output of `assempde`

.

`B`

— Dirichlet nullspace

sparse matrix

Dirichlet nullspace, returned as a sparse matrix. See Algorithms. Typically, `B`

is
the output of `assempde`

.

`ud`

— Dirichlet vector

vector

Dirichlet vector, returned as a vector. See Algorithms. Typically, `ud`

is
the output of `assempde`

.

`M`

— Mass matrix

sparse matrix | full matrix

Mass matrix. specified as a sparse matrix or a full matrix. See Elliptic Equations.

To obtain the input matrices for `pdeeig`

, `hyperbolic`

or `parabolic`

,
run both `assema`

and `assempde`

:

[Kc,Fc,B,ud] = assempde(model,c,a,f); [~,M,~] = assema(model,0,d,f);

**Note**

Create the `M`

matrix using `assema`

with `d`

,
not `a`

, as the argument before `f`

.

**Data Types: **`double`

**Complex Number Support: **Yes

`rtol`

— Relative tolerance for ODE solver

`1e-3`

(default) | positive real

Relative tolerance for ODE solver, specified as a positive real.

**Example: **`2e-4`

**Data Types: **`double`

`atol`

— Absolute tolerance for ODE solver

`1e-6`

(default) | positive real

Absolute tolerance for ODE solver, specified as a positive real.

**Example: **`2e-7`

**Data Types: **`double`

## Output Arguments

`u`

— PDE solution

matrix

PDE solution, returned as a matrix. The matrix is `Np`

**N*-by-`T`

,
where `Np`

is the number of nodes in the mesh, *N* is
the number of equations in the PDE (*N* = 1 for a
scalar PDE), and `T`

is the number of solution times,
meaning the length of `tlist`

. The solution matrix
has the following structure.

The first

`Np`

elements of each column in`u`

represent the solution of equation 1, then next`Np`

elements represent the solution of equation 2, etc. The solution`u`

is the value at the corresponding node in the mesh.Column

`i`

of`u`

represents the solution at time`tlist`

`(i)`

.

To obtain the solution at an arbitrary point in the geometry,
use `pdeInterpolant`

.

To plot the solution, use `pdeplot`

for
2-D geometry, or see 3-D Solution and Gradient Plots with MATLAB Functions.

## Algorithms

### Reducing Parabolic Equations to Elliptic Equations

`parabolic`

internally calls `assema`

,
`assemb`

, and `assempde`

to create finite element
matrices corresponding to the problem. It calls `ode15s`

to solve the resulting system of ordinary differential
equations.

Partial Differential Equation Toolbox™ solves equations of the form

$$m\frac{{\partial}^{2}u}{\partial {t}^{2}}+d\frac{\partial u}{\partial t}-\nabla \xb7\left(c\nabla u\right)+au=f$$

When the *m* coefficient is 0, but *d* is not, the
documentation refers to the equation as *parabolic*, whether or not it
is mathematically in parabolic form.

A parabolic problem is to solve the equation

$$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\nabla u\right)+au=f\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\Omega $$

with the initial condition

*u*(**x**,0) =
*u*_{0}(**x**) for
**x**∊Ω

where **x** represents a 2-D or 3-D point and there are
boundary conditions of the same kind as for the elliptic equation on ∂Ω.

The heat equation reads

$$\rho C\frac{\partial u}{\partial t}-\nabla \text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(k\nabla u\right)+h\left(u-{u}_{\infty}\right)=f$$

in the presence of distributed heat loss to the surroundings. *ρ* is the
density, *C* is the thermal capacity, *k* is the thermal
conductivity, *h* is the film coefficient,
*u*_{∞} is the ambient temperature, and
*f* is the heat source.

For time-independent coefficients, the steady-state solution of the equation is the solution to the standard elliptic equation

–∇ · (*c*∇*u*) + *au* =
*f*.

Assuming a mesh on Ω and *t* ≥ 0, expand the solution to the PDE (as a
function of **x**) in the Finite Element Method basis:

$$u(x,t)={\displaystyle \sum _{i}{U}_{i}(t){\varphi}_{i}(x)}$$

Plugging the expansion into the PDE, multiplying with a test function
*ϕ _{j}*, integrating over Ω, and applying
Green's formula and the boundary conditions yield

$$\begin{array}{l}{\displaystyle \sum _{i}{\displaystyle \underset{\Omega}{\int}d{\varphi}_{j}{\varphi}_{i}\text{\hspace{0.17em}}\frac{d{U}_{i}\left(t\right)}{dt}}}\text{\hspace{0.17em}}dx+{\displaystyle \sum _{i}\left({\displaystyle \underset{\Omega}{\int}\left(\nabla {\varphi}_{j}\cdot \left(c\nabla {\varphi}_{i}\right)+a{\varphi}_{j}{\varphi}_{i}\right)\text{\hspace{0.17em}}dx+{\displaystyle \underset{\partial \Omega}{\int}q{\varphi}_{j}{\varphi}_{i}\text{\hspace{0.17em}}ds}}\right){U}_{i}(t)}\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}={\displaystyle \underset{\Omega}{\int}f{\varphi}_{j}\text{\hspace{0.17em}}dx}+{\displaystyle \underset{\partial \Omega}{\int}g{\varphi}_{j}\text{\hspace{0.17em}}ds}\text{\hspace{1em}}\forall j\end{array}$$

In matrix notation, we have to solve the *linear, large* and
*sparse* ODE system

$$M\frac{dU}{dt}+KU=F$$

This method is traditionally called *method of lines*
semidiscretization.

Solving the ODE with the initial value

*U _{i}*(0) =

*u*

_{0}(

**x**

*)*

_{i}yields the solution to the PDE at each node **x*** _{i}* and time

*t*. Note that

*K*and

*F*are the stiffness matrix and the right-hand side of the elliptic problem

–∇ · (*c*∇*u*) + *au* =
*f* in Ω

with the original boundary conditions, while *M* is just the mass matrix
of the problem

–∇ · (0∇*u*) + *du* = 0 in Ω.

When the Dirichlet conditions are time dependent, *F * contains
contributions from time derivatives of *h* and **r**. These derivatives are evaluated by finite differences of the
user-specified data.

The ODE system is ill conditioned. Explicit time integrators are forced by stability
requirements to very short time steps while implicit solvers can be expensive since they
solve an elliptic problem at every time step. The numerical integration of the ODE system is
performed by the MATLAB^{®} ODE Suite functions, which are efficient for this class of problems. The time
step is controlled to satisfy a tolerance on the error, and factorizations of coefficient
matrices are performed only when necessary. When coefficients are time dependent, the
necessity of reevaluating and refactorizing the matrices each time step may still make the
solution time consuming, although `parabolic`

reevaluates only that which
varies with time. In certain cases a time-dependent Dirichlet matrix **h**(*t*) may cause the error control to fail, even if the
problem is mathematically sound and the solution *u*(*t*)
is smooth. This can happen because the ODE integrator looks only at the reduced solution
*v* with *u* = *Bv* +
*ud*. As **h** changes, the pivoting scheme
employed for numerical stability may change the elimination order from one step to the next.
This means that *B*, *v*, and *ud* all
change discontinuously, although *u* itself does not.

## Version History

**Introduced before R2006a**

### R2016a: Not recommended

`parabolic`

is not recommended. Use `solvepde`

instead. There are no plans to remove
`parabolic`

.

### R2012b: Coefficients of parabolic PDEs as functions of the solution and its gradient

You can now solve parabolic equations whose coefficients depend on the solution
*u* or on the gradient of *u*.

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