Soft Computing https://doi.org/10.1007/s00500-020-05181-3 METHODOLOGIES AND APPLICATION On a new weight tri-diagonal iterative method and its applications D. Yambangwai 1 · W. Cholamjiak 1 · T. Thianwan 1 · H. Dutta 2 © Springer-Verlag GmbH Germany, part of Springer Nature 2020 Abstract In this article, a new weight tri-diagonal iterative method in solving system of linear equation is proposed and its convergence is discussed. The set of methods in solving linear system generated by high-order scheme for solving one-dimensional Poisson equation and one-dimensional heat equation with periodic boundary are presented. The numerical experiments shows that the proposed method demonstrates a better performance compared with weight Jacobi, successive-over relaxation and alternating group explicit methods. Keywords Iterative method · Convergence · High-order scheme 1 Introduction With the increasing availability of powerful computing machine and the improvement of numerical technique, finite difference methods play an important role in the solution of mathematical applications. The most frequently used finite difference schemes (FDS) are based on the second-order central difference scheme (CDS). In numerous applications, the solving of tri-diagonal linear systems is required when second-order CDS is reported in a number of publications because of their simplicity and robustness. However, the high-order difference scheme (HDS) is required in many problems of industrial and scientific inter- est if we seek for the smoothness of the solution due to the discontinuities or high gradient regions existed in the solu- tions, the better steady state convergence, the better provable Communicated by V. Loia. B D. Yambangwai damrongsak.ya@up.ac.th W. Cholamjiak watcharaporn.ch@up.ac.th T. Thianwan tanakit.th@up.ac.th H. Dutta duttah@gauhati.ac.in 1 Department of Mathematics, University of Phayao, Phayao 56000, Thailand 2 Department of Mathematics, Gauhati University, Guwahati 781014, India convergence properties and the better efficiency. On the other hand, HDS is often complicated to understand and code and costly run with the same mesh as compared with the lower- order method and less robust. Normally, the stencil of HDS becomes wider with increas- ing order of accuracy. For a high-order method of traditional type, the standard centered discretization of order p, the sten- cil is p + 1 points wide. And, the discretization of differential equations with periodic solution using p order standard cen- tered FDS leads to the system of linear algebraic equation Au = f , (1) where u is the variable, f is the source term and A is real, symmetric matrix of nearly block p-diagonal form (El- Mikkawy 2002; Sogabe 2008). It can be easily realized that direct methods are not appropriate for solving large number of equations in a system when the coefficient matrix A is large and sparse (Yousef 2019). Thus, traditionally iterative methods have been used to solve such problem and presented in solving the solution of such linear systems. One of them is the basic iterative method (Samarskii 2001 take in the form M  u k +1 − u k  + Au k = f , (2) when matrix A is split by matrix M as A = M + ( A − M ). Due to the simplicity of coding and understanding, the iter- ative method take in the form (2), is still widely used for the solution of the linear system. However, this basic iterative method is more efficient if a weight parameter for accel- erating the iteration process is predominantly applied. The 123