IX. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam

Abstract

1. Introduction.— 1·0. The object of this paper is to develop methods where by the differential equations of physics may be applied more freely than hitherto in the approximate form of difference equations to problems concerning irregular bodies. Though very different in method, it is in purpose a continuation of a former paper by the author, on a “Freehand Graphic Way of Determining Stream Lines and Equipotentials” (‘Phil. Mag.,’February, 1908; also ‘Proc. Physical Soc.,’ London, vol. xxi.). And all that was there said, as to the need for new methods, may be taken to apply here also. In brief, analytical methods are the foundation of the whole subject, and in practice they are the most accurate when they will work, but in the integration of partial equations, with reference to irregular-shaped boundaries, their field of application is very limited.

On the truncation error in the solution of Laplaces equation by finite differences

“…We are now ready to estimate lu -u"l. To this end we wri te, using (1) and (2) and from (7) it follows that 00 R3"S:. K :6 n-3 = Kh 2 /2.…”

Richardson Extrapolation

“…As mentioned before, this algorithm was introduced and discussed by L. F. Richardson in 1911and 1927, see Richardson (1911, 1927. It should also be mentioned here that L. F. Richardson called this procedure "deferred approach to the limit".…”

“…We are interested in the case where the method based on (5.55)-(5.57) is used in a combination with both the Marchuk-Strang splitting procedure, see Marchuk (1968Marchuk ( , 1980Marchuk ( , 1982Marchuk ( , 1986Marchuk ( , 1988) and Strang (1968) and the Richardson Extrapolation, see Faragó, Havasi and Zlatev (2010) and Richardson (1911Richardson ( , 1927. This combination is a new numerical method and we shall study some stability properties of the combined numerical method both in the general case when the Runge-Kutta methods is defined by (5.56)-(5.57) and in some particular cases.…”

Implicit Virtual Boundary Method for Moving Boundary Problems on Non-Staggered Cartesian Patch Grids

“…In 1910, Richardson [42] proposed an explicit three-time-level scheme for solving transient heat conduction. The unsteady term was discretized with the standard central difference method such that the scheme is second-order accurate in the time coordinate.…”