TY - JOUR
N2 - This paper presents the concept of using algorithms for reducing the dimensions of finite-difference equations of two-dimensional (2D) problems, for second-order partial differential equations. Solutions are predicted as two-variable functions over the rectangular domain, which are periodic with respect to each variable and which repeat outside the domain. Novel finite-difference operators, of both the first and second orders, are developed for such functions. These operators relate the value of derivatives at each point to the values of the function at all points distributed uniformly over the function domain. Aâ€¯specific feature of the novel operators follows from the arrangement of the function values as well as the values of derivatives, which are rectangular matrices instead of vectors. This significantly reduces the dimensions of the finite-difference operators to the numbers of points in each direction of the 2D area. The finite-difference equations are created exemplary elliptic equations. An original iterative algorithm is proposed for reducing the process of solving finite-difference equations to the multiplication of matrices.
L1 - http://www.journals.pan.pl/Content/118365/PDF/29_D1535-1541_01751_Bpast.No.68-6_29.12.20_OK.pdf
L2 - http://www.journals.pan.pl/Content/118365
PY - 2020
IS - No. 6
EP - 1541
DO - 10.24425/bpasts.2020.135387
KW - second-order partial differential equations
KW - 2D finite-difference operators
KW - finite-difference equations
KW - iterative algorithms
A1 - Sobczyk, T.
VL - 68
DA - 31.12.2020
T1 - Finite-Difference Operators for 2D problems
SP - 1535
UR - http://www.journals.pan.pl/dlibra/publication/edition/118365
T2 - Bulletin of the Polish Academy of Sciences Technical Sciences
ER -