The Methods of Numerical Time Integration of One-dimensional Linear Equations and Their Inherited Errors

Abstract

For solving one-dimensional linear partial differential equations numerically, several methods are studied on the accuracy, the stability and the nherited errors of solutions. First, the time extrapolation scheme is discussed, and the 3 time-level method (the usual centered difference) and one of the 4-level methods are concluded to give satisfactory results. In particular, the latter is numerically stable even for the equation of frictional type. Second, concerning the formulas of approximating the space derivative by finite difference, the 3 gridpoint and the 5 gridpoint methods are examined. The differential equations treated involves the advection term, where the advecting flow is spatially variable. Namely, the term is consisted of the product of two space-dependent quantities, which produces new waves by their interaction. It is shown that the usual computing method, i.e., the “3-3pt” method (3 levels for time and 3 points for space difference) gives arise the small scale errors, which turn out to be the ruggedness of the resulted pattern. On the contrary, the “3-5pt” method is stable and much accurate for the equation excluding the friction term and the “4-5pt” method for any kind of equations. The fictitious waves in small scale are extraordinarily small by these methods, The measures whereby one can expect the occurrence of the error are presented. They are derived from the difference between the differential and the difference equations, and they are related to the degree of the spatial undulation of the quantity concerned. Finally, using the techniques discussed, some sample calculations of 500 hour forecast are performed with respect to the patterns moving on a one-dimensional ringwise grid, and the numerical solutions are compared with the analytical solutions.