Abstract
According to Godunov theorem for numerical calculations of advection equations, there exist no high-er-order schemes with constant positive difference coefficients in a family of polynomial schemes with an accuracy exceeding the first-order. In case of advection-diffusion equations, so far there have been not found stable schemes with positive difference coefficients in a family of numerical schemes exceeding the second-order accuracy. We propose a third-order computational scheme for numerical fluxes to guarantee the non-negative difference coefficients of resulting finite difference equations for advection-diffusion equations. The present scheme is optimized so as to minimize truncation errors for the numerical fluxes while fulfilling the positivity condition of the difference coefficients which are variable depending on the local Courant number and diffusion number. The feature of the present optimized scheme consists in keeping the third-order accuracy anywhere without any numerical flux limiter by using the same stencil number as convemtional third-order shemes such as KAWAMURA and UTOPIA schemes. We extend the present method into multi-dimensional equations. Numerical experiments for linear and nonlinear advection-diffusion equations were performed and the present scheme’s applicability to nonlinear Burger’s equation was confirmed.
Highlights
In numerical calculations of advection equations and advection-diffusion equations appearing frequently in scientific and engineering fields, there is a trend of tradeoff relationship between numerical accuracy and numerical stability
The feature of the present optimized scheme consists in keeping the third-order accuracy anywhere without any numerical flux limiter by using the same stencil number as convemtional third-order shemes such as KAWAMURA and UTOPIA schemes
We proposed a third-order polynomial scheme for numerical fluxes to guarantee the positive difference coefficients of resulting finite difference equations for
Summary
In numerical calculations of advection equations and advection-diffusion equations appearing frequently in scientific and engineering fields, there is a trend of tradeoff relationship between numerical accuracy and numerical stability. In case of the conventional high- order polynomial schemes such as QUICK [1], UTOPIA [1] and KAWAMURA [2] schemes, at least one of those difference coefficients is negative even for advection- diffusion equations, and those higher-order schemes tend to bring forth unstable solutions due to unphysical oscillations, especially around a location where a steep gradient in the solutions exists To cope with this numerical oscillation problems, nonlinear monotonicity preserving schemes such as the FRAM technique [3] and the TVD schemes [4] using a numerical flux limiter have been proposed. By involving the properties of the exact solutions of linear and nonlinear advection-diffusion equations into a numerical scheme, we constructed the numerical schemes ANO [5,6] and COLE [6,7] with monotonocity preserving properties for unsteady linear and nonlinear equations, respectively
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