A compact fourth-order spatial discretisation applied to the Navier-Stokes equations

Linjing Zhou, Steven Armfield


Modern direct and large eddy simulation of turbulent and transition flows requires accurate solution of the Navier--Stokes equations. High accuracy is achieved using a high order discretisation. The standard high order approach for local methods, such as finite-difference or finite-volume, produces large computational molecules and thus introduces complexity in the boundary treatment and parallelisation. Existing compact schemes need to invert a matrix system, which increases the computational cost, and are restricted to application on non-uniform grids. The fourth-order compact scheme proposed here iteratively applies a low order compact method to achieve higher accuracy. The scheme allows for a simple application of boundary conditions, can be applied on a non-uniform grid and allows a standard parallelisation approach to be used. The scheme is implemented and tested in an unsteady finite-difference heat equation solver and benchmarked against the analytical solution to validate the order of accuracy. It is also included in a full fractional-step Navier-Stokes solver and validated for the lid-driven cavity problem.


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DOI: http://dx.doi.org/10.21914/anziamj.v56i0.9422

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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.