4.4 Source terms

Most ``conservation laws" include source terms (ex: relativistic hydrodynamic equations)

$$\frac{\partial {\bf u}}{\partial t} + \frac{\partial {\bf f... ...m} {\bf f} = {\bf f}({\bf u}), {\bf s} = {\bf s}({\bf u})$$ (119)

Two basic ways of handling source terms:

1. Unsplit methods: a single finite difference formula advances the full equation over one time step
   $$\begin{eqnarray*} {\bf u}_j^{n+1} = {\bf u}_j^n - \frac{\Delta t}{\Delta x} \lef... ...2}} - \hat{\bf f}_{j-\frac{1}{2}}\right) + \Delta t {\bf s} \end{eqnarray*}$$
   
   
   
   
   
   

   This algorithm can be improved by introducing succesive sub-steps to perform the time update (ex: predictor-corrector, Shu & Osher's conservative high order Runge-Kutta schemes)

2. Fractional step or splitting methods: split the equation into different pieces (transport + sources) and apply appropriate methods for each piece independently

   ${\bf u}^{n+1} = {\cal L}_s^{\Delta t} {\cal L}_f^{\Delta t} {\bf u}^n$ (Godunov splitting, first order)

   1. First step (PDE), ${\cal L}_f^{\Delta t}$: $$\frac{\partial {\bf u}}{\partial t} + \frac{\partial {\bf f}({\bf u})}{\partial x} = 0 \hspace{0.4cm} \rightarrow \hspace{0.4cm} {\bf u}^*$$

   2. Second step (ODE), ${\cal L}_s^{\Delta t}$: $$\frac{\partial {\bf u}^*}{\partial t} = {\bf s}({\bf u}^*) \hspace{0.4cm} \rightarrow \hspace{0.4cm} {\bf u}^{n+1}$$

   to get second order accuracy (assuming each independent method is second order) $\rightarrow$ Strang splitting: ${\bf u}^{n+1} = {\cal L}_s^{\Delta t/2} {\cal L}_f^{\Delta t} {\cal L}_s^{\Delta t/2} {\bf u}^n$

   ${\bf u}^n \hspace{0.3cm} \longrightarrow \hspace{0.3cm} {\bf u}^* \hspace{0.3c... ....3cm} {\bf u}^{**} \hspace{0.3cm} \longrightarrow \hspace{0.3cm} {\bf u}^{n+1}$

   $\Delta t/2$ $\Delta t$ $\Delta t/2$

   source transport source

Stiff source terms: model phenomena which

1. occur on much faster time scales than $\Delta t$
2. act over much smaller spatial scales than $\Delta x$

$\rightarrow$ lead to numerical difficulties

• Example: combustion

  the chemical reactions (or nuclear reactions in stars) occur on much faster time scales than the gas flow $\rightarrow$ waves may propagate at nonphysical speeds

  Model problem: two chemical species, ``unburnt gas" and "burnt gas"

  unburnt gas $\longrightarrow$ burnt gas $k(T)$: reaction rate

  $k(T)$

  $$\begin{eqnarray*} k(T)= \left\{ \begin{array}{cc} 1/\tau & T \ge T_0 \\ 0 & T < T_0 \end{array}\right. \end{eqnarray*}$$
  
  
  
  
  
  

  $T_0$: ignition temperature

  $\tau$: time scale of the chemical reaction

  Equations:

  $$\begin{eqnarray*} (\rho)_t + (\rho u)_x &=& 0 \\ (\rho u)_t + (\rho u^2 + p)_x ... ...+ (u(E+p))_x &=& 0 \\ (\rho Z)_t + (\rho u Z)_x &=& -k(T)\rho Z \end{eqnarray*}$$
  
  
  
  
  
  

  $$E=\frac{1}{2}\rho u^2 + \frac{p}{\gamma-1} + q_0\rho Z$$

  $Z$: mass fraction of the unburnt gas

  $\gamma$: adiabatic exponent

  $q_0$: heat release