Monotonic unbounded schemes transformer (MUST): an approach to remove undershoots and overshoots in family of unbounded schemes using finite-volume method

• Published in: International journal of numerical methods for heat & fluid flow Vol. 34; no. 11; pp. 4049 - 4084
• Main Authors: Yap, Y.F., Chai, J.C.
• Format: Journal Article
• Language: English
• Published: Bradford Emerald Publishing Limited 30-09-2024; Emerald Group Publishing Limited
• Subjects: Boundary conditions; Convection; Dependent variables; Discretization; Finite volume method; Peclet number; Reynolds number; Bounded higher-order spatial differencing scheme; Finite-volume method
• ISSN: 0961-5539; 0961-5539; 1758-6585
• DOI: 10.1108/HFF-04-2024-0293

Purpose
This paper presents a Monotonic Unbounded Schemes Transformer (MUST) approach to bound/monotonize (remove undershoots and overshoots) unbounded spatial differencing schemes automatically, and naturally. Automatically means the approach (1) captures the critical cell Peclet number when an unbounded scheme starts to produce physically unrealistic solution automatically, and (2) removes the undershoots and overshoots as part of the formulation without requiring human interventions. Naturally implies, all the terms in the discretization equation of the unbounded spatial differencing scheme are retained.
Design/methodology/approach
The authors do not formulate new higher-order scheme. MUST transforms an unbounded higher-order scheme into a bounded higher-order scheme.
Findings
The solutions obtained with MUST are identical to those without MUST when the cell Peclet number is smaller than the critical cell Peclet number. For cell Peclet numbers larger than the critical cell Peclet numbers, MUST sets the nodal values to the limiter value which can be derived for the problem at-hand. The authors propose a way to derive the limiter value. The authors tested MUST on the central differencing scheme, the second-order upwind differencing scheme and the QUICK differencing scheme. In all cases tested, MUST is able to (1) capture the critical cell Peclet numbers; the exact locations when overshoots and undershoots occur, and (2) limit the nodal value to the value of the limiter values. These are achieved by retaining all the discretization terms of the respective differencing schemes naturally and accomplished automatically as part of the discretization process. The authors demonstrated MUST using one-dimensional problems. Results for a two-dimensional convection–diffusion problem are shown in Appendix to show generality of MUST.
Originality/value
The authors present an original approach to convert any unbounded scheme to bounded scheme while retaining all the terms in the original discretization equation.