I am trying to solve a non-linear second order PDE as given below.
$$\partial_{t}u+u\partial_{x}u=\partial_{xx}u+u$$
with periodic boundary condition: $$u(x,t)=u(x+2\pi,t)$$ and initial condition: $$u(x,0)=u_{0}(x)$$
I am using a fractional step method to solve this problem as shown below.
- 1st step: Non-linear part $$\partial_{t}u=-u\partial_{x}u$$
- 2nd step: Linear part $$\partial_{t}u=\partial_{xx}u+u$$ using Fourier discretization in space.
But I do not get reasonable results and there are oscillations.
Any suggestions for any other method for solving such PDEs?