# Finite Difference Approximation for the Laplacian in 2D that produces a nonsymmetric matrix

Consider the following PDE

\begin{align} -\Delta u &= f \ \ \text{en} \ \ (0,1)\times (0,1) \label{P1} \\ u &= 0 \ \ \text{en} \ \partial ((0,1)\times (0,1)) \label{P2} \end{align}

if we use 5-point finite difference approximations we get a system $$$$Ax = b \label{Sist_Planteamiento}$$$$

where the matrix $$A$$ is symmetric.

Questions:

1. Do you know a finite difference approximation that produces a system
$$$$Ax = b$$$$ where $$A$$ is nonsymmetric?

2. Do you know books or papers where I can study systems like the system in question 1?

• I guess you could also use non-equally weighted (but still centered) finite difference approximations: $f'' \approx [w_1f(x+h) - w_2f(x) + w_3(x-h)]/h^2$ where $w_2 = w_1+w_3$ and $w_1\neq w_3$. As long as the discretization is consistent and stable, it would be convergent. But I don't know why anyone would want such a scheme aside from academic curiosity. Jul 25 at 17:57
• @ChennaK I want to test a preconditioner in the system $A^{T}Ax=A^{T}b$. Jul 25 at 17:58