Simple Implementation of 1D TVD Scheme for 2D Triangular Finite Volume
Abstract
In this paper a simple implementation of 1D TVD scheme to 2D triangular grid was proposed and sufficient condition for oscillation free solution of advection equation were found using monotone advective K-approximation. The approach was implemented to Superbee and Smart limiter and compared to Barth and Jespersen (BJ) scheme by using the schemes to well known classic case of advection of step and double step of scalar properties. Result indicated that all of the computed solutions are monotone and, apart from highly diffusive first order solution, they show similar level accuracy for test cases. Superbee limiter gives the best performance and follow by Smart limiter and BJ scheme.
##Keywords:##
Monotone, Limiter, Triangular grid, Finite volume.
Published
Nov 30, 2024
How to Cite
TASRI, Adek.
Simple Implementation of 1D TVD Scheme for 2D Triangular Finite Volume.
Journal of Ocean, Mechanical and Aerospace -science and engineering-, [S.l.], v. 68, n. 3, p. 169-174, nov. 2024.
ISSN 2527-6085.
Available at: <https://isomase.org/Journals/index.php/jomase/article/view/524>. Date accessed: 19 mar. 2025.
doi: http://dx.doi.org/10.36842/jomase.v68i3.524.
References
[1] Leonard, B.P. (1980). The quick algorithm: A uniformly third-order finite volume method for highly convective flow. In K. Morgan et al. (Eds.), Computer Methods in Fluid. Penthech Press.
[2] Jawahar, P. & Kamath, H. (2000). A high-resolution procedure for Euler and Navier-Stokes computations on unstructured grids. Journal of Computational Physics, 164(1), 165–203.
[3] Jameson, A. (1994). Analysis and design of numerical schemes for gas dynamics: I. Artificial diffusion, upwind biasing, limiters, and their effect on accuracy and multigrid convergence. International Journal of Computational Fluid Dynamics, 3(2), 172–218.
[4] Jameson, A. & Mavripilis, D. (1985). Finite volume solution of the two-dimensional Euler equations on a regular triangular mesh. AIAA Paper 85-0435.
[5] Harten, A. (1983). High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 49(3), 357–393.
[6] Van Leer, B. (1974). Towards the ultimate conservation difference scheme. II. Monotonicity and conservation combined in a second-order scheme. Journal of Computational Physics, 14(4), 361–370.
[7] Boris, J.P. & Book, D.L. (1973). Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works. Journal of Computational Physics, 11(1), 38–69.
[8] Sweby, P.K. (1984). High-resolution schemes using flux limiters for hyperbolic conservation laws. SIAM Journal on Numerical Analysis, 21(5), 995–1011.
[9] Berzins, M. & Ware, J.M. (1995). Positive cell-centered finite volume discretization methods for hyperbolic equations on irregular meshes. Applied Numerical Mathematics, 16(5–6), 417–438.
[10] Tasri, A. (2005). Accuracy of nominally second-order unstructured grid CFD codes (PhD thesis). University of Newcastle, UK.
[11] Barth, T.J. & Jespersen, D.C. (1989). The design and application of upwind schemes on unstructured meshes. AIAA Paper 89-0366.
[12] Venkatakrishnan, V. (1993). On the accuracy of limiters and convergence to steady-state solutions. AIAA Paper 93-0880.
[13] Aftosmis, M & Tavares, T.S. (1994). The behavior of linear reconstruction on unstructured meshes. Wright Laboratory Report.
[14] Trefethen, L.N. (1982). Group velocity in finite difference schemes. SIAM Review, 24(2), 113–136.
[15] Leonard, B.P. (1979). A stable and accurate convective modeling procedure based on quadratic upstream interpolation. Computer Methods in Applied Mechanics and Engineering, 19(1), 59–98.
[16] Roe, P.L. (1981). The use of the Riemann problem in finite difference schemes. In Lecture Notes in Physics (Vol. 141, pp. 354–359).
[17] Roe, P L. (1985). Some contributions to the modeling of discontinuous flow. In Lectures in Applied Mathematics (Vol. 22, pp. 163–193).
[18] Gaskell, P.H. & Lau, A.K.C. (1988). Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm. International Journal for Numerical Methods in Fluids, 8(6), 617–641.
[19] Tasri, A. (2002). TVD method for unstructured grid (Internal report). University of Newcastle, UK.
[20] Darwish, M.S. & Moukalled, F. (2003). TVD schemes for unstructured grids. International Journal of Heat and Mass Transfer, 46(4), 599–611.
[21] Bruner, C. & Walter, R. (1995). Parallelization of the Euler equations on unstructured grids. AIAA Paper 97-894.
[22] Lin, S.Y. & Yu, T.M. (1993). Upwind finite volume methods with triangular meshes for conservation laws. Journal of Computational Physics, 107(2), 324–337.
[23] Wilder, P. & Fotia, G. (2002). A positive spatial advection scheme on unstructured meshes for tracer transport. Journal of Computational and Applied Mathematics, 140(1–2), 809–821.
[24] Van Leer, B. (1979). Towards the ultimate conservation difference scheme. V. A second-order sequel to Godunov’s method. Journal of Computational Physics, 32(1), 101–118.
[25] Cho, H.K., Lee, H.D., Park, I.K. & Jeong, J.J. (2010). Implementation of second-order upwind methods in a semi-implicit two-phase flow code on unstructured meshes. Annals of Nuclear Energy, 37(5), 606–614.
[2] Jawahar, P. & Kamath, H. (2000). A high-resolution procedure for Euler and Navier-Stokes computations on unstructured grids. Journal of Computational Physics, 164(1), 165–203.
[3] Jameson, A. (1994). Analysis and design of numerical schemes for gas dynamics: I. Artificial diffusion, upwind biasing, limiters, and their effect on accuracy and multigrid convergence. International Journal of Computational Fluid Dynamics, 3(2), 172–218.
[4] Jameson, A. & Mavripilis, D. (1985). Finite volume solution of the two-dimensional Euler equations on a regular triangular mesh. AIAA Paper 85-0435.
[5] Harten, A. (1983). High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 49(3), 357–393.
[6] Van Leer, B. (1974). Towards the ultimate conservation difference scheme. II. Monotonicity and conservation combined in a second-order scheme. Journal of Computational Physics, 14(4), 361–370.
[7] Boris, J.P. & Book, D.L. (1973). Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works. Journal of Computational Physics, 11(1), 38–69.
[8] Sweby, P.K. (1984). High-resolution schemes using flux limiters for hyperbolic conservation laws. SIAM Journal on Numerical Analysis, 21(5), 995–1011.
[9] Berzins, M. & Ware, J.M. (1995). Positive cell-centered finite volume discretization methods for hyperbolic equations on irregular meshes. Applied Numerical Mathematics, 16(5–6), 417–438.
[10] Tasri, A. (2005). Accuracy of nominally second-order unstructured grid CFD codes (PhD thesis). University of Newcastle, UK.
[11] Barth, T.J. & Jespersen, D.C. (1989). The design and application of upwind schemes on unstructured meshes. AIAA Paper 89-0366.
[12] Venkatakrishnan, V. (1993). On the accuracy of limiters and convergence to steady-state solutions. AIAA Paper 93-0880.
[13] Aftosmis, M & Tavares, T.S. (1994). The behavior of linear reconstruction on unstructured meshes. Wright Laboratory Report.
[14] Trefethen, L.N. (1982). Group velocity in finite difference schemes. SIAM Review, 24(2), 113–136.
[15] Leonard, B.P. (1979). A stable and accurate convective modeling procedure based on quadratic upstream interpolation. Computer Methods in Applied Mechanics and Engineering, 19(1), 59–98.
[16] Roe, P.L. (1981). The use of the Riemann problem in finite difference schemes. In Lecture Notes in Physics (Vol. 141, pp. 354–359).
[17] Roe, P L. (1985). Some contributions to the modeling of discontinuous flow. In Lectures in Applied Mathematics (Vol. 22, pp. 163–193).
[18] Gaskell, P.H. & Lau, A.K.C. (1988). Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm. International Journal for Numerical Methods in Fluids, 8(6), 617–641.
[19] Tasri, A. (2002). TVD method for unstructured grid (Internal report). University of Newcastle, UK.
[20] Darwish, M.S. & Moukalled, F. (2003). TVD schemes for unstructured grids. International Journal of Heat and Mass Transfer, 46(4), 599–611.
[21] Bruner, C. & Walter, R. (1995). Parallelization of the Euler equations on unstructured grids. AIAA Paper 97-894.
[22] Lin, S.Y. & Yu, T.M. (1993). Upwind finite volume methods with triangular meshes for conservation laws. Journal of Computational Physics, 107(2), 324–337.
[23] Wilder, P. & Fotia, G. (2002). A positive spatial advection scheme on unstructured meshes for tracer transport. Journal of Computational and Applied Mathematics, 140(1–2), 809–821.
[24] Van Leer, B. (1979). Towards the ultimate conservation difference scheme. V. A second-order sequel to Godunov’s method. Journal of Computational Physics, 32(1), 101–118.
[25] Cho, H.K., Lee, H.D., Park, I.K. & Jeong, J.J. (2010). Implementation of second-order upwind methods in a semi-implicit two-phase flow code on unstructured meshes. Annals of Nuclear Energy, 37(5), 606–614.
