The nonlinear modified equation approach to analyzing finite difference schemes

Adaptive finite difference solutions of Liouville equations in computational uncertainty quantification

“…While the adaptive mesh approach based upon equidistribution attempts to minimize the error in regions of strong gradients and local extrema, another possibility in reducing numerical error is to attempt to find a mesh distribution which equidistributes or minimizes the local truncation error or its estimate [37,38,39,40,41]. In this current paper, the approach we will use in this class of mesh adaptation methods is similar in concept to that presented by Klopfer and McRae [38].…”

“…In this current paper, the approach we will use in this class of mesh adaptation methods is similar in concept to that presented by Klopfer and McRae [38]. In their work a one-dimensional shock tube problem was solved using the MacCormack finite difference scheme with mesh clustering based upon the modified differential equation.…”

Grid Generation for Fluid Mechanics Computations

“…Further optimization would be expected when special techniques are developed for special situations. The moving finite-element method described by Miller & Miller (1981) and the modified-equation technique proposed by Klopfer & McRae (1981) are both optimal in their respective measures, and they are restricted to distinct types of solution procedures. For a class of convection-diffusion problems, the transformation into simpler diffusion equations was simultaneously considered by Piva et al (1982) and .…”

Minimization of the Truncation Error by Grid Adaptation

“…Another class of methods is based on equidistribution or minimization of the local truncation error or its estimate [7][8][9][10]. In [7] the error estimate obtained by using a finite difference approximation of the leading truncation error term is equidistributed by the grid point redistribution.…”

“…In [7] the error estimate obtained by using a finite difference approximation of the leading truncation error term is equidistributed by the grid point redistribution. Klopfer and McRae [8] solve a one-dimensional shock-tube problem with the explicit predictor-corrector scheme of MacCormack on a grid dynamically adapted to the solution. The error estimate is the leading truncation error term of the differential equations transformed to the computational coordinates.…”