Abstract
The main focus of this study was to develop a numerical scheme with new expressions for interface flux approximations based on the upwind approach in the finite volume method. Our new proposed numerical scheme is unconditionally stable with second-order accuracy in both space and time. The method is based on the second-order formulation for the temporal approximation, and an upwind approach of the finite volume method is used for spatial interface approximation. Some numerical experiments have been conducted to illustrate the performance of the new numerical scheme for a convection–diffusion problem. For the phenomena of convection dominance and diffusion dominance, we developed a comparative study of this new upwind finite volume method with an existing upwind form and central difference scheme of the finite volume method. The modified numerical scheme shows highly accurate results as compared to both numerical schemes.
Highlights
Partial differential equations (PDEs) have a vital role in describing different phenomena in real life
The finite volume method is a special kind of numerical approach which is widely used for solving convection–diffusion problems in computational fluid dynamics [23,24,25,26,27]
With the consideration of convection dominance or flow direction, new expressions for interface approximations have been obtained for the directional flow phenomenon, and using those approximations a numerical scheme was developed with second-order accuracy in both space and time
Summary
Partial differential equations (PDEs) have a vital role in describing different phenomena in real life. The comparison of finite difference and finite volume method for the numerical approximation of the convection–diffusion equation was studied in the papers [19,20,21,22]. Numerical simulations of non-Newtonian fluids were investigated using the finite volume formulation [35], where the approximation of the convection process is managed by using the QUICK differences scheme. In the past, based on spatial interface approximations, several finite volume numerical schemes were developed to solve the convection–diffusion problem. Crank–Nicolson formulation is used for the approximation along the temporal direction, and using these new interface approximations, a numerical scheme was developed for the numerical approximation of convection–diffusion problem.
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