How can i solve this first order system of differential equations

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I am trying to solve $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu\frac{\partial^{2}u}{\partial x^{2}}\\ u(0,t)=0\nonumber \\ u(1,t)=0\nonumber \\ u(x,0)=\sin(\pi x)\\ 0<x<1,\; t>0$$ I take $u_n(x,t)$ as an approximation to $u(x,t)$ $$u_{n}(x,t)=\sum_{n=-4}^{2^j-1}\phi_{n}(x)c_{n}(t),\quad j\in \mathbb{N}$$ Here $\phi_n(x)$ are basis functions and i need to find the coeffiicients $c_n$. If i take the weight functions as $\phi_k(x)$ then weighted integral form of my problem will be as follows:

$$\int_{0}^{1}\phi_{k}(x)\sum_{n}\phi_{n}(x)c_{n}^{'}(t)dx+\int_{0}^{1}\phi_{k}(x)\left(\sum_{n}\phi_{n}(x)c_{n}(t)\right)\left(\sum_{l}\phi'_{l}(x)c_{l}(t)\right)dx\\-\int_{0}^{1}\phi_{k}(x)\nu\sum_{n}\phi''_{n}(x)c_{n}(t)dx=0$$ Now, set $$\delta_{k,n}=\int_{0}^{1}\phi_{k}\phi_{n}dx,\quad\Omega_{k,n}^{0,2}=\int_{0}^{1}\phi_{k}\phi''_{n},\quad\Omega_{k,n,l}^{0,0,1}=\int_{0}^{1}\phi_{k}\phi_{n}\phi'_{l}dx$$ Therefore discretized version of my problem becomes. $$\sum_{n=-4}^{0}c'_{n}(t)\delta_{k,n}+\sum_{n=-4}^{0}c_{n}(t)\sum_{l=-4}^{0}c_{l}(t)\Omega_{k,n,l}^{0,0,1}-\nu\sum_{n=-4}^{0}c_{n}(t)\Omega_{k,n}^{0,2}=0$$ $$\left\{ \mathbf{c}_{n}'(t)\right\} \left[\delta_{k,n}\right]+\left\{ \mathbf{c}_{n}(t)\right\} \left[\mathbf{c}_{l}(t)\Omega_{k,n,l}^{0,0,1}\right]-\nu\left\{ \mathbf{c}_{n}(t)\right\} \left[\Omega_{k,n}^{0,2}\right]=0\\ $$ Here i know the values of $\delta_{k,n}$, $\Omega_{k,n,l}^{0,0,1}$, $\Omega_{k,n}^{0,2}$ and $\nu$. Also $-4 \leq k,n,l \leq 2^j-1$. Approximating $c'_{n}(t),c_{n}(t)$ by $$c'_{n}(t)=\frac{c_{n}^{i+1}-c_{n}^{i}}{h}$$ $$c_{n}(t)=\frac{c_{n}^{i+1}+c_{n}^{i}}{2}$$ I get $$\mathbf c_{n}^{i+1}\left(\frac{\delta_{k,n}}{h}+\frac{\mathbf c_{l}\Omega_{k,n,l}^{0,0,1}}{2}-\nu\frac{\Omega_{k,n}^{0,2}}{2}\right)=\mathbf c_{n}^{i}\left(\frac{\delta_{k,n}}{h}-\frac{\mathbf c_{l}\Omega_{k,n,l}^{0,0,1}}{2}+\nu\frac{\Omega_{k,n}^{0,2}}{2}\right)$$ I know $c_n^i$ for $i=1$ from initial condition and i want to find $c_{n}^{i+1}$

If $c_l$ not get into the job i could find the $c_n$. But i am confused with $c_l$.

Answer 1

What you describe as your time discretization is called the Crank-Nicolson scheme. For nonlinear differential equations it leads, as you have observed, to a nonlinear algebraic system that needs to be solved at each time step. The typical approach is to solve it with a Newton method -- in your case, that requires to solve a nonlinear system in 5 variables. Any introductory book on numerical methods will explain several different methods of doing that.

Equations such as yours are pretty straightforward and without great difficulty. Rather than implementing a numerical scheme yourself, may I suggest you simply describe the system of differential equations to Maple, Mathematica or Matlab and let them solve it to essentially any accuracy you desire?

Answer 2

As it looks to me your formulation of the discrete system is incomplete. You need to

1. state your equation at each (grid?) point associated with $n$
2. put back the summation over $l$ (grid) points in your last equation
3. assign a time index $i+1$ or $i$ to your $c_l$s

In this way, you get a nonlinear (if you take $c_l^{n+1}$) or linear (if you take $c_l^{n}$) system of equations.

Depending on your discretization, the resulting system will have no / one / or multiple solutions. If it has one, you can use standard linear solvers or a Newton scheme to solve it. If not, you need more considerations and maybe a remodelling of your system.

You should also try, to write your equation system in matrix/vector form, which I find easier to handle than the formulation with the summations and indices.