A New Finite-Difference Solution-Adaptive Method

• Published in: Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences physical, and engineering sciences, 1992-12, Vol.341 (1662), p.373-410
• Main Authors: Hawken, D. F., Hansen, J. S., Gottlieb, J. J.
• Format: Article
• Language: English
• Subjects: Boundary conditions; Classical and quantum physics: mechanics and fields; Classical mechanics of continuous media: general mathematical aspects; Coefficients; Elastic waves; Entropy; Error rates; Exact sciences and technology; Flow velocity; Jacobians; Physics; Reynolds number; Shock tubes; Shock waves
• ISSN: 1364-503X; 0962-8428; 1471-2962; 2054-0299
• DOI: 10.1098/rsta.1992.0109

A new moving finite difference (MFD) method has been developed for solving hyperbolic partial differential equations and is compared with the moving finite element (MFD) method of K. Miller and R. N. Miller. These methods involve the adaptive movement of nodes so as to reduce the number of nodes needed to solve a problem; they are applicable to the solution of non-stationary flow problems that contain moving regions of rapid change in the flow variables, surrounded by regions of relatively smooth variation. Both methods solve simultaneously for the flow variables and the node locations at each time-step, and they move the nodes so as to minimize an ‘error’ measure that contains a function of the time derivatives of the solution. This error measure is manipulated to obtain a matrix equation for node velocities. Both methods make use of penalty functions to prevent node crossing. The penalty functions result in extra terms in the matrix equation that promote node repulsion by becoming large when node separation becomes small. Extensive work applying the MFE and MFD methods to one-dimensional gasdynamic problems has been conducted to evaluate their performance. The test problems include Burgers’ equation, ideal viscid planar flow within a shock-tube, propagation of shock and rarefaction waves through area changes in ducts, and viscous transition through a contact surface and a shock.