# parabolic

Solve parabolic PDE problem

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.

## Syntax

• u = parabolic(u0,tlist,model,c,a,f,d)
example
• u = parabolic(u0,tlist,b,p,e,t,c,a,f,d)
example
• u = parabolic(u0,tlist,Kc,Fc,B,ud,M)
example
• u = parabolic(___,rtol)
• u = parabolic(___,rtol,atol)
• u = parabolic(___,'Stats','off')

## Description

example

u = parabolic(u0,tlist,model,c,a,f,d) produces the solution to the FEM formulation of the scalar PDE problem$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\nabla u\right)+au=f,$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.

example

u = parabolic(u0,tlist,b,p,e,t,c,a,f,d) solves the problem using boundary conditions b and finite element mesh specified in [p,e,t].

example

u = parabolic(u0,tlist,Kc,Fc,B,ud,M) solves the problem based on finite element matrices that encode the equation, mesh, and boundary conditions.
u = parabolic(___,rtol) and u = parabolic(___,rtol,atol), for any of the previous input arguments, modify the solution process by passing to the ODE solver a relative tolerance rtol, and optionally an absolute tolerance atol.
u = parabolic(___,'Stats','off'), for any of the previous input arguments, turns off the display of internal ODE solver statistics during the solution process.

## Examples

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### Parabolic Equation

Solve the parabolic equation

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 on all edges.

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

Generate a relatively fine mesh.

generateMesh(model,'Hmax',0.02); 

Set the initial condition to have on the disk and 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); 
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(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

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 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 on the disk and 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,'Face',[1,4,6],'u',0); 

The initial temperature distribution in the block has the form

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

Suppose the thermal conductivity of the block leads to a coefficient value of 1. The values of the other coefficients in this problem are , , and .

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); 
38 successful steps 0 failed attempts 78 function evaluations 1 partial derivatives 10 LU decompositions 77 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' caxis([umin umax]); subplot(2,2,2) pdeplot3D(model,'colormapdata',u(:,11)) colorbar off view(125,22) title 't = 10' caxis([umin umax]); subplot(2,2,3) pdeplot3D(model,'colormapdata',u(:,26)) colorbar off view(125,22) title 't = 25' caxis([umin umax]); subplot(2,2,4) pdeplot3D(model,'colormapdata',u(:,41)) colorbar off view(125,22) title 't = 40' caxis([umin umax]); 

## Input Arguments

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### u0 — Initial conditionvector | text expression

Initial condition, specified as a scalar, vector of nodal values, or text expression. 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. For details, see Initial Conditions.

• 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 as

char('x.^2 + 5*cos(x.*y)',... 'tanh(x.*y)./(1+z.^2)')

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

Data Types: double | char
Complex Number Support: Yes

### tlist — Solution timesreal 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 modelPDEModel object

PDE model, specified as a PDEModel object.

Example: model = createpde(1)

### c — PDE coefficientscalar or matrix | character array | coefficient function

PDE coefficient, specified as a scalar or matrix, as a character array, or as a 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

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

You can specifyc in various ways, detailed in c Coefficient for Systems. See also Scalar PDE Coefficients, Specify Scalar PDE Coefficients in String Form, Specify 2-D Scalar Coefficients in Function Form, Specify 3-D PDE Coefficients in Function Form, and Coefficients for Systems of PDEs.

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

Data Types: double | char | function_handle
Complex Number Support: Yes

### a — PDE coefficientscalar or matrix | character array | coefficient function

PDE coefficient, specified as a scalar or matrix, as a character array, or as a 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

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

You can specifya in various ways, detailed in a or d Coefficient for Systems. See also Scalar PDE Coefficients, Specify Scalar PDE Coefficients in String Form, Specify 2-D Scalar Coefficients in Function Form, Specify 3-D PDE Coefficients in Function Form, and Coefficients for Systems of PDEs.

Example: 2*eye(3)

Data Types: double | char | function_handle
Complex Number Support: Yes

### f — PDE coefficientscalar or matrix | character array | coefficient function

PDE coefficient, specified as a scalar or matrix, as a character array, or as a 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

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

You can specifyf in various ways, detailed in f Coefficient for Systems. See also Scalar PDE Coefficients, Specify Scalar PDE Coefficients in String Form, Specify 2-D Scalar Coefficients in Function Form, Specify 3-D PDE Coefficients in Function Form, and Coefficients for Systems of PDEs.

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

Data Types: double | char | function_handle
Complex Number Support: Yes

### d — PDE coefficientscalar or matrix | character array | coefficient function

PDE coefficient, specified as a scalar or matrix, as a character array, or as a 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

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

You can specifyd in various ways, detailed in a or d Coefficient for Systems. See also Scalar PDE Coefficients, Specify Scalar PDE Coefficients in String Form, Specify 2-D Scalar Coefficients in Function Form, Specify 3-D PDE Coefficients in Function Form, and Coefficients for Systems of PDEs.

Example: 2*eye(3)

Data Types: double | char | function_handle
Complex Number Support: Yes

### b — Boundary conditionsboundary matrix | boundary file

Boundary conditions, specified as a boundary matrix or boundary file. Pass a boundary file as a function handle or as a string naming the file.

For more information on boundary conditions, see Forms of Boundary Condition Specification.

Example: b = 'circleb1' or equivalently b = @circleb1

Data Types: double | char | function_handle

### p — Mesh nodesoutput of initmesh | output of meshToPet

Mesh nodes, specified as the output of initmesh or meshToPet. For the structure of a p matrix, see Mesh Data for [p,e,t] Triples: 2-D and Mesh Data for [p,e,t] Triples: 3-D.

Example: [p,e,t] = initmesh(g)

Data Types: double

### e — Mesh edgesoutput of initmesh | output of meshToPet

Mesh edges, specified as the output of initmesh or meshToPet. For the structure of e, see Mesh Data for [p,e,t] Triples: 2-D and Mesh Data for [p,e,t] Triples: 3-D.

Example: [p,e,t] = initmesh(g)

Data Types: double

### t — Mesh elementsoutput of initmesh | output of meshToPet

Mesh elements, specified as the output of initmesh or meshToPet. Mesh elements are the triangles or tetrahedra that form the finite element mesh. For the structure of a t matrix, see Mesh Data for [p,e,t] Triples: 2-D and Mesh Data for [p,e,t] Triples: 3-D.

Example: [p,e,t] = initmesh(g)

Data Types: double

### Kc — Stiffness matrixsparse 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 vectorvector

Load vector, specified as a vector. See Elliptic Equations. Typically, Fc is the output of assempde.

### B — Dirichlet nullspacesparse matrix

Dirichlet nullspace, returned as a sparse matrix. See Algorithms. Typically, B is the output of assempde.

### ud — Dirichlet vectorvector

Dirichlet vector, returned as a vector. See Algorithms. Typically, ud is the output of assempde.

### M — Mass matrixsparse 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 solver1e-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 solver1e-6 (default) | positive real

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

Example: 2e-7

Data Types: double

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example:

### 'Stats' — Display ODE solver statistics'on' (default) | 'off'

Display ODE solver statistics, specified as 'on' or 'off'. Suppress the display by setting Stats to 'off'.

Example: x = parabolic(u0,tlist,model,c,a,f,d,'Stats','off')

Data Types: char

## Output Arguments

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### u — PDE solutionmatrix

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 number of elements 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 Plot 3-D Solutions.

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### Algorithms

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. For details, see Parabolic Equations.