Partial Differential Equations (PDEs)

Many physical phenomena around us are governed by advection-diffusion-reaction partial differential equations.

In the scalar case (single equation), it can be written as follows:

\[\frac{\partial q}{\partial t}+\frac{\partial f(q)}{\partial x}=s(x, t, q)+\frac{\partial}{\partial x}\left(\alpha \frac{\partial q}{\partial x}\right)\]
  • \(q=q(x, t)\) is the unknown dependent variable and a function of two independent variables \(x\) and \(t\).
  • \(f(q)\) is the flux.
  • \(s(x,t,q)\) is the source term
  • \(\frac{\partial}{\partial x}\left(\alpha \frac{\partial q}{\partial x}\right)\) is the diffusion term, which is second-order differential operator.
  • \(\alpha=\alpha(x,t,q)\) is the diffusion coefficient.

Linear Equation

The linear advection-diffusion-reaction partial differential equation, obtained by setting the flux function as linear function of \(q\), \(f(q)=\lambda q\), and the source function as linear function of \(q\), \(s(q)=\beta q\), can be written as follows:

\begin{equation} \frac{\partial q}{\partial t}+\lambda \frac{\partial q}{\partial x}=\beta q+\alpha \frac{\partial^2 q}{\partial x} \label{eq:le} \end{equation}

with \(\lambda\), \(\beta\), and \(\alpha\) of constants.

Assuming that there is no advection, \(\partial_x q=0\), and no diffusion, \(\alpha=0\), Eq. \eqref{eq:le} becomes an ordinary differential equation (ODE) as follows:

\[\frac{d q}{d t}=\beta q(x,t) \equiv s(t,q)\]

Initial Value Problem (IVP)

By setting source coefficient and diffusion coefficient as zero, the linear advection equation (LAE) can be written as follows:

\begin{equation} \frac{\partial q}{\partial t}+\lambda \frac{\partial q}{\partial x}=0 \label{eq:lae} \end{equation}

with characteristic speed \((\lambda)\).

Because of initial value problem, the spatial domain is infinite \((-\infty< 0< \infty)\), i.e., no spatial boundaries, and time \((t>0)\) is positive. An initial condition at time \((t=0)\) is provided as follows:

\[q(x,0)=h(x)\]

Characteristic Curves

Eq. \eqref{eq:lae} can be expressed by characteristic curves, which is function, \(x(t)\), defined in the \(x\)-\(t\) half-plane of independent variables \((x, t)\) as follows:

\[\frac{d x(t)}{d t}=\lambda\]

satisfying the IVP \(x(0)=x_0\). This solution can be easily obtained as follows:

\[x(t)=x_0+\lambda t\]

where \(x_0\) is foot of the characteristic and the characteristic speed \((\lambda)\) determines the slope of the characteristics. Therefore, in case of linear case with constant \(\lambda\), characteristics are all parallel to each other.

In order to obtain the solution \((q)\) of original PDE equation, Eq. \eqref{eq:lae}, consider that the time-rate of change of \(q\) along a characteristic curve \(x=x(t)\) as follows:

\[\frac{dq}{dt}=\frac{\partial q}{\partial t}\frac{d t}{d x}+\frac{\partial q}{\partial x}\frac{d x}{d t}=\partial_{t}q+\lambda \partial_{x}q=0\]

As a result, the linear advection equation \(\partial_{t}q+\lambda \partial_{x}q=0\) (PDE) becomes the ODE \(d_{t}q=0\) along the characteristics, \(x=x_{0}+\lambda t\). This states that \(q\) is constant along the characteristics \(x=x_{0}+\lambda t\). In other words, the value of \(q(x,t)\) at arbitrary point \((x,t)\) is equal to the value of \(q\) at the foot \(x_0\) as follows:

\begin{equation} q(x,t)=q(x_0,t)=h(x_0)=h(x-\lambda t) \label{eq:charac} \end{equation}

Proof

To see that the function \(q(x,t)\) in Eq. \eqref{eq:charac} is the exact solution of the original IVP, Eq. \eqref{eq:lae}, assume that the initial condition \(h(x)\) is differentiable (i.e. smooth function).

\[\frac{\partial q}{\partial t}=\frac{\partial h}{\partial (x-\lambda t)}\times \frac{\partial (x-\lambda t)}{\partial t} \equiv h'\times(-\lambda)\] \[\frac{\partial q}{\partial x}=\frac{\partial h}{\partial (x-\lambda t)}\times \frac{\partial (x-\lambda t)}{\partial x} \equiv h'\times(\lambda)\]

Substituting these partial derivatives into the original PDE,

\[\frac{\partial q}{\partial t}+\lambda \frac{\partial q}{\partial x}=-\lambda h'+\lambda h'=0.\]

Therefore, the function \(q(x,t)=h(x-\lambda t)\) is the solution of the original PDE.

Riemann Problem

The Riemann problem for the linear advection equation is the special IVP including the discontinuity solution as follows:

\[\frac{\partial q}{\partial t}+\lambda \frac{\partial q}{\partial x}=0\]

with initial condition

\[q(x,0)=h(x) = \begin{cases}q_L & \text { if } x<0 \\ q_R & \text { if } x>0\end{cases}\]

At the point \(x=0\), the solution is not defined, i.e., singularity.

The solution is as follows:

\[q(x,t)=h(x-\lambda t) = \begin{cases}q_L & \text { if } x-\lambda t<0 \\ q_R & \text { if } x-\lambda t>0\end{cases}\]

Initial-Boundary Value Problem (IBVP)

In most practical problem, the boundaries are present and then have to be defined in a finite spatial domain \(x\in[a,b]\) as follows:

\[q= \begin{cases} q(a,t)=b_{L}(t) \\ q(b,t)=b_{R}(t) \end{cases}\]

In addition to IC, the boundary conditions are need for computation.

For \(\lambda >0\), the solution in domain \(x\in[a,b]\) depends on the IC and the left BC because the values at left BC are constantly propagated to the right as follows:

\[q(x,t)=\begin{cases} h(x-\lambda t) & \text{ if } x-\lambda t >a \\ b_L(t+\frac{a-x}{\lambda}) & \text{ if } x-\lambda t <a \end{cases}\]

Generalization

Linear Advection-Reaction Equation

In addition ot LAE, the reaction term can be added in equation as follows:

\[\frac{\partial q}{\partial t}+\lambda \frac{\partial q}{\partial x}=\beta q.\]

The exact solution can be expressed by

\[q(x,t)=h(x-\lambda t)\exp(\beta t),\]

where \(q\) represents a wave traveling at speed \(\lambda\), attenuated in time by the factor \(\exp(\beta t)\). For stability of the PDE, it is assumed that \(\beta \le 0\). In case of \(\beta >0\), \(q\) would increase unboundedly in time, i.e., unstable.

Advection with Variable Coefficient

In case that the coefficient is not constant, \(\lambda\), but variable, \(\lambda (x,t)\), the equation can be expressed by

\[\frac{\partial q}{\partial t}+\lambda (x,t) \frac{\partial q}{\partial x}=0.\]

The characteristic speed \(\lambda (x,t)\) is no longer constant but a function of space and time. Thus, the associated characteristics are no longer straight lines and no longer parallel to each other although the solution is constant along the characteristics.

Linear Advection-Reaction in 3D

The linear advection-reaction equation in three space dimensions can be expressed by

\[\frac{\partial q}{\partial t}+\lambda_1\frac{\partial q}{\partial x}+\lambda_2\frac{\partial q}{\partial y}+\lambda_3\frac{\partial q}{\partial z}=\beta q,\]

with initial condition, \(q(x,y,z,0)=h(x,y,z)\), where \(\lambda_1, \lambda_2\), and \(\lambda_3\) denote the characteristic speeds in the \(x, y\), and \(z\) direction, respectively.

The exact solution can be expressed by

\[q(x,y,z,t)=h(x-\lambda_1 t, y-\lambda_2 t,z-\lambda_3 t)\exp(\beta t).\]