Boundary-fitted coordinate systems for numerical solution of partial differential equations—A review

Review of some adaptive node-movement techniques in finite-element and finite-difference solutions of partial differential equations

pdf/sparse-identification-of-truncation-errors-1u216gy8aq.pdf

Sparse identification of truncation errors

Hyperbolic heat conduction with radiation is considered in a finite absorbing and emitting slab for the cases of a constant and a pulsed heat flux boundary condition. For small optical thicknesses, the wave nature of hyperbolic heat conduction gives rise to a thermal front propagating from the back surface which results from the heating of the back surface by radiation through the medium. MacCormack's explicit predictor-corrector scheme is used to solve the problem, and the results from the hyperbolic solutions are compared to those obtained from the parabolic equation.

Hyperbolic heat conduction with radiation in an absorbing and emitting medium

The construction of dynamically-adaptive curvilinear coordinate systems based on numerical grid generation and the use thereof in the numerical solution of partial differential equations is surveyed, and correlations are made among the various approaches. These adaptive grids are coupled with the physical solution being done on the grid so that the grid points continually move in the course of the solution in order to resolve developing gradients, or higher variations, in the solution. Particular attention is given to systems using elliptic grid generation based on variational principles. It is noted that dynamic grid adaption can remove the oscillations common when strong gradients occur on fixed grids, and that it appears that when the grid adapts to the solution most numerical solution algorithms work well. Particular applications in computational fluid dynamics and heat transfer are noted.

A survey of dynamically-adaptive grids in the numerical solution of partial differential equations

The nonlinear modified equation approach is taken in this paper to analyze the generalized Lax-Wendroff explicit scheme approximation to the unsteady one- and two-dimensional equations of gas dynamics. Three important applications of the method are demonstrated. The nonlinear modified equation analysis is used to (1) generate higher order accurate schemes, (2) obtain more accurate estimates of the discretization error for nonlinear systems of partial differential equations, and (3) generate an adaptive mesh procedure for the unsteady gas dynamic equations. Results are obtained for all three areas. For the adaptive mesh procedure, mesh point requirements for equal resolution of discontinuities were reduced by a factor of five for a 1-D shock tube problem solved by the explicit MacCormack scheme.

The nonlinear modified equation approach to analyzing finite difference schemes

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The nonlinear modified equation approach to analyzing finite difference schemes