Symmetric Versus Nonsymmetric Differencing
Wayne R. Dyksen, John R. Rice · SIAM Journal on Scientific and Statistical Computing · 1985
Consider the self-adjoint elliptic problem $(pu_x )_x + (qu_y )_y + ru = f$ with Dirichlet boundary conditions on the unit square. This problem is symmetric in the sense that if the data is symmetric then so is the solution. The straightforward finite difference discretization has one expand the derivatives and apply differences to $pu_{xx} + p_x u_x + \cdots $. Alternatively there are symmetric discretizations which are attractive intuitively and which are usually recommended. We have observed that symmetric discretizations are sometimes much less accurate; a simple analysis is made to compare the expected behavior of the two discretizations. Data from a simplified model problem confirms the expectations that nonsymmetric differences are more accurate than symmetric differences much more often than vice versa. We conclude for elliptic problems that unless it is known that u varies much more rapidly than p and q, one should use nonsymmetric differences.