dev.nm.analysis.differentialequation.pde.finitedifference.parabolic.dim1.convectiondiffusionequation

Class ConvectionDiffusionEquation1D

java.lang.Object

dev.nm.analysis.differentialequation.pde.finitedifference.parabolic.dim1.convectiondiffusionequation.ConvectionDiffusionEquation1D

All Implemented Interfaces:

PDE

public class ConvectionDiffusionEquation1D
extends Object
implements PDE

The convection–diffusion equation is a combination of the diffusion and convection (advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Depending on context, the same equation can be called the advection–diffusion equation, drift–diffusion equation, Smoluchowski equation (after Marian Smoluchowski), or (generic) scalar transport equation.

A one-dimensional general convection-diffusion equation is of the following form. \[ \frac{\partial u}{\partial t} = \sigma(t,x) \frac{\partial^2 u}{\partial x^2} - \mu(t,x) \frac{\partial u}{\partial x} + R(t,x), \] The initial condition has this form: \[ u(0,x) = f(x), ~0 < x < a \] The mixed boundary conditions have this form: \[ (1-c_1)u(t,0) - c_1\frac{\partial u}{\partial x}(t,0) = g_1(t), \\ (1-c_2)u(t,a) + c_2\frac{\partial u}{\partial x}(t,a) = g_2(t) \] where \(0 \le c_k \le 1, k = 1, 2\).

See Also:

Wikipedia: Convection-diffusion equation

Constructor Summary

Constructors

Constructor and Description
ConvectionDiffusionEquation1D(BivariateRealFunction sigma, BivariateRealFunction mu, BivariateRealFunction R, double a, double T, UnivariateRealFunction f, double c1, UnivariateRealFunction g1, double c2, UnivariateRealFunction g2) Constructs a convection-diffusion equation problem.

Method Summary

Modifier and Type	Method and Description
double	a() Gets the size of the one-dimensional space, that is, the range of x, (0 < x < a).
double	c1() Gets the coefficient \(c_{1}\) in the mixed boundary condition at the boundary x = 0. \[ (1-c_1)u(t,0) - c_1\frac{\partial u}{\partial x}(t,0) = g_1(t) \] where \(0 \leq c_1 \leq 1\).
double	c2() Gets the coefficient \(c_{2}\) in the mixed boundary condition at the boundary x = a. \[ (1-c_2)u(t,a) + c_2\frac{\partial u}{\partial x}(t,a) = g_2(t) \] where \(0 \leq c_2 \leq 1\).
double	f(double x) Gets the initial condition of u at a given position x.
double	g1(double t) The value of the linear combination of \(u\) and \(\frac{\partial u}{\partial x}\) at the boundary \(x = 0\) at a given time \(t\). \[ (1-c_1)u(t,0) - c_1\frac{\partial u}{\partial x}(t,0) = g_1(t) \]
double	g2(double t) The value of the linear combination of \(u\) and \(\frac{\partial u}{\partial x}\) at the boundary \(x = a\) at the given time \(t\). \[ (1-c_2)u(t,a) + c_2\frac{\partial u}{\partial x}(t,a) = g_2(t) \]
double	mu(double t, double x) Gets the convection coefficient at a given time t and a position x.
double	R(double t, double x) Gets the source (or sink) value at a given time t and a position x.
double	sigma(double t, double x) Gets the diffusion coefficient at a given time t and a position x.
double	T() Gets the time period of interest, that is, the range of t, (0 < t < T).

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

ConvectionDiffusionEquation1D

public ConvectionDiffusionEquation1D(BivariateRealFunction sigma,
                                     BivariateRealFunction mu,
                                     BivariateRealFunction R,
                                     double a,
                                     double T,
                                     UnivariateRealFunction f,
                                     double c1,
                                     UnivariateRealFunction g1,
                                     double c2,
                                     UnivariateRealFunction g2)

Constructs a convection-diffusion equation problem.

Parameters:

sigma - the diffusion coefficient (or diffusivity) \(\sigma(t,x)\)

mu - the convection coefficient \(\mu(t,x)\)

R - the sources (or sinks) R(t,x)

a - the region of interest (0, a)

T - the time period of interest (0, T)

f - the initial condition of u, i.e., u(0,x)

c1 - the coefficient in the mixed boundary condition at x = 0

g1 - the mixed boundary condition at x = 0

c2 - the coefficient in the mixed boundary condition at x = a

g2 - the mixed boundary condition at x = a

Method Detail

sigma

public double sigma(double t,
                    double x)

Gets the diffusion coefficient at a given time t and a position x.

Parameters:

t - a time

x - a position

Returns:

the diffusion coefficient \(\sigma(t,x)\)

mu

public double mu(double t,
                 double x)

Gets the convection coefficient at a given time t and a position x.

Parameters:

t - a time

x - a position

Returns:

the convection coefficient \(\mu(t,x)\)

R

public double R(double t,
                double x)

Gets the source (or sink) value at a given time t and a position x.

Parameters:

t - a time

x - a position

Returns:

the source (or sink) R(t,x)

a

public double a()

Gets the size of the one-dimensional space, that is, the range of x, (0 < x < a).

Returns:

the size of the space

T

public double T()

Gets the time period of interest, that is, the range of t, (0 < t < T).

Returns:

the time period of interest

f

public double f(double x)

Gets the initial condition of u at a given position x.

Parameters:

x - a position

Returns:

u(0, x)

c1

public double c1()

Gets the coefficient \(c_{1}\) in the mixed boundary condition at the boundary x = 0. \[ (1-c_1)u(t,0) - c_1\frac{\partial u}{\partial x}(t,0) = g_1(t) \] where \(0 \leq c_1 \leq 1\). If \(c_1 = 0\), this is the Dirichlet boundary condition. If \(c_1 = 1\), this is the Neumann boundary condition.

Returns:

the coefficient \(c_{1}\)

g1

public double g1(double t)

The value of the linear combination of \(u\) and \(\frac{\partial u}{\partial x}\) at the boundary \(x = 0\) at a given time \(t\). \[ (1-c_1)u(t,0) - c_1\frac{\partial u}{\partial x}(t,0) = g_1(t) \]

Parameters:

t - a time

Returns:

\(g_1(t)\)

c2

public double c2()

Gets the coefficient \(c_{2}\) in the mixed boundary condition at the boundary x = a. \[ (1-c_2)u(t,a) + c_2\frac{\partial u}{\partial x}(t,a) = g_2(t) \] where \(0 \leq c_2 \leq 1\). If \(c_2 = 0\), this is the Dirichlet boundary condition. If \(c_2 = 1\), this is the Neumann boundary condition.

Returns:

the coefficient \(c_{2}\)

g2

public double g2(double t)

The value of the linear combination of \(u\) and \(\frac{\partial u}{\partial x}\) at the boundary \(x = a\) at the given time \(t\). \[ (1-c_2)u(t,a) + c_2\frac{\partial u}{\partial x}(t,a) = g_2(t) \]

Parameters:

t - a time

Returns:

\(g_2(t)\)