Higher-order conservative discretizations on arbitrarily varying non-uniform grids

A. Arun Govind Neelan; Raimund Bürger; Manoj T. Nair; Samala Rathan

doi:10.29393/ppudec-11ocne50011

##article.authors##

A. Arun Govind Neelan Department of Mechanical Engineering, Indian Institute of Technology-Madras, Chennai, 600036, Tamil Nadu, India.
Raimund Bürger Universidad de Concepción, Facultad de Ciencias Físicas y Matemática, Departamento de Ingeniería Matemática. Concepción, Chile.
Manoj T. Nair Department of Aerospace Engineering, Indian Institute of Space Science and Technology, Thiruvananthapuram, 695547, Kerala, India.
Samala Rathan Department of Humanities Sciences, Indian Institute of Petroleum & Energy, Visakhapatnam, 530003, Andhra Pradesh, India.

DOI:

https://doi.org/10.29393/ppudec-11ocne50011

Keywords:

Finite volume method, finite difference scheme, conservative discretization, WENO, ENO

Resumen

Conservative discretizations of transport equations are based on integral formulationsthat include the finite volume method (FVM) and conservative finite difference methods (CFDMs). The FVM is used by most fluid dynamics simulation packages and require ssmoothly shifting grids for better convergence. This motivates the study of the order of accuracy and rate of convergence of the FVM on non-uniform grids. It is difficult to do such an analysis of the FVM on an unstructured grid; however, the FVM is reduced to a CFDM on a Cartesian grid. The effect of the order of accuracy and the rate of convergence of higher-order CFDMs on arbitrarily varying grids is therefore investigated. It is shown that higher-order conservative discretizations on arbitrarily varying non-uniform grids are not even first-order accurate in theoretical and numerical test cases. This property is verified by a symbolic computation procedure. If the grid is replaced by a gradually varying grid, it is shown that conservative discretizations yield a better rate of convergence. In this situation a rate of convergence between one and the theoretical maximum can be achieved in dependence of the grid stretch/- contraction ratio. Numerical examples including the linear convectionm-diffusiom equation, the lid-driven cavity problem, and the Taylor-Green vortex problem are presented.

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