# Method to linearize highly nonlinear partial differential equation

I have a set of coupled pdes which I want to solve using finite-difference, of which one is nonlinear. The three linear pdes for quantities $$T_f$$, $$T_s$$ and $$c$$ are convection-diffusion-reaction-like equations. The nonlinear equation for $$\alpha$$ is of the Arrhenius type:

$$\frac{\partial \alpha}{\partial t}=A_fexp\left(\frac{-E_a}{RT_s}\right)(1-\alpha)^{2/3}\left(1-\frac{B_f exp\left(\frac{-D}{T_f}\right)}{cRT_f}\right)$$,

of which $$A_f$$, $$E_a$$, $$R$$, $$B_f$$ and $$D$$ are constants.

The problem has to be solved implicitly, preferably using the Crank-Nicolson method. What is the way to go here? My guess is that the equation above has to be linearized iteratively. Is the (modified) Newton method applicable here? What would the implementation look like?

Your system is of the form $$y'=f(t,y)$$, with $$y \in \mathbb{R}^m$$ your state vector.

Applying Crank-Nicolson to that scheme leads to the following equation for the time step from $$t_n$$ to $$t_{n+1}=t_n + \Delta t$$:

$$y_{n+1} = y_{n} + \frac{\Delta t}{2} \left( f(y_n) + f(y_{n+1}) \right)$$

This corresponds to the following nonlinear problem: $$O = g(y_{n+1})$$ with: $$g(y_{n+1}) = y_{n+1} - \frac{\Delta t}{2} f(y_{n+1}) - \underbrace{ ( y_{n} + \frac{\Delta t}{2} f(y_n) )}_{R}$$ The term $$R$$ does not depend on $$y_{n+1}$$.

The traditional way si to apply a Newton method, which linearises the problem iteratively to solve the nonlinear problem. Starting from the an initial guess $$y_{n+1}^0$$ (simplest guess is $$y_n$$), the $$k$$-th Newton iteration reads:

$$y_{n+1}^k = y_{n+1}^{k-1} - (\partial_y g)^{-1} g(y_{n+1}^{k-1})$$

For a full Newton method, the Jacobian is evaluated at $$y_{n+1}^k$$, and is therefore updated at each iteration. Very often for ODEs, it is sufficient to perform quasi-Newton iterations, where this Jacobian is only updated if the Newton converges poorly, e.g. if $$||y_{n+1}^{k+1} - y_{n+1}^{k}||$$ does not decrease quickly enough from one iteration to the next. The Newton is ensured to converge for a sufficiently small time step.

The other approach you suggested is to linearise $$f$$ directly in the Crank-Nicolson scheme: $$f(t_{n+1}, y_{n+1}) \approx f(y_{n}) + (\partial_y f)(y_n) (y_{n+1}-y_{n})$$

In that case, a single Newton iteration suffices, because $$g$$ is then linear: $$\left(\mathbb{I} - \dfrac{\Delta t}{2}\partial_y f(y_n)\right) y_{n+1} = \left( \mathbb{I} - \dfrac{\Delta t}{2} \partial_y f(y_n)\right) y_n + \Delta t f(y_n)$$

However, that approach is typically much less robust and stable in the case of strongly nonlinear terms, because the linearisation if too crude of an approximation.

Remarks:

• maybe an explicit algorithm can work very well for your problem if this nonlinear term you speak of is not stiff. If you have stiffness issues due to other terms (e.g. diffusive processes), try explicit integrators with extended stability domains (e.g. ROCK4).

• in all cases, try to use existing softwares to integrate your system. Coding in Fortran ? Try Radau5, Dopri5. Coding in Matlab, try ode45, ode15s, ode32tb. Coding in Python, try the integrators from Scipy's solve_ivp package.

• Laurent, thank you for your elaborate answer. If I understand correctly, each inner iteration a system of equations is solved of the form $y^k_{n+1}=y^{k-1}_{n+1}-J^{-1}(y^{k-1}_{n+1})g(y^{k-1}_{n+1})$. The Jacobian $J$ is a matrix of partial derivatives of the four pdes, but with respect to which variables exactly? Regarding your remarks: I've tried solving the Arrhenius equation explicitly, but this results in too tight time step restrictions. I will look into your suggestion. Dec 21, 2021 at 11:03