A Positivity-preserving Finite Volume Scheme Based on Second-order Scheme

Abstract

Abstract: Based on a second-order accurate linear scheme, a normal flux is reconstructed to obtain a nonlinear finite volume scheme with a two-point flux discrete stencil on tetrahedral meshes. It is suitable for discontinuous and anisotropic diffusion coefficient problems, and can be generalized to general polyhedral meshes. It is unnecessary to assume that auxiliary unknowns are non-negative, and avoids artificial processing of "setting negative to be zero" in calculating auxiliary unknowns. Moreover, it is proved that the linearized scheme at each nonlinear iteration step satisfies strong positivity-preserving, i.e., as the source term and boundary condition are non-negative, non-zero solution of the scheme is strictly greater than zero. Numerical tests verify that the scheme has second-order accuracy and is strong positivity-preserving.

Key words: diffusion equations, tetrahedral meshes, finite volume scheme, strong positivity-preserving

CLC Number:

O241.82

ZHAO Fei, SHENG Zhiqiang, YUAN Guangwei. A Positivity-preserving Finite Volume Scheme Based on Second-order Scheme[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2020, 37(4): 379-392.

References

[1] YU C X, FAN Z F, LIU J, et al. Modeling of shell-mixing into central hotspot in inertial confinement fusion implosion[J]. Chinese J Comput Phys, 2017, 34(4):379-386.
[2] SHEN Y S, LI K B, SHI X M, et al. Study on disposing high-level transuranic waste aste in a fusion fission reactor[J]. Chinese J Comput Phys, 2017, 34(2):142-148.
[3] YUAN G W, HANG X D, SHENG Z Q, et al. Progress in numerical methods for radiation diffusion equations[J]. Chinese J Comput Phys, 2009, 26(4):475-500.
[4] LI D Y, SHUI H S, TANG M J. On the finite difference scheme of two-dimensional parabolic equation in a non-rectangular mesh[J]. J Numer Methods Comput Appl, 1980, 1:217-224.
[5] SHASHKOV M, STEINBERG S. Support-operator finite-difference algorithms for general elliptic problems[J]. J Comput Phys, 1995, 118:131-151.
[6] AAVATSMARK I. An introduction to multipoint flux approximations for quadrilateral grids[J]. Comput Geosci, 2002, 6:405-432.
[7] CAO F J, YAO Y Z. Conservative positivity-preserving algorithm for Kershaw scheme of anisotropic diffusion problem[J]. Chinese J Comput Phys, 2017, 34(3):283-293.
[8] YAO Y Z, YUAN G W. Enforcing positivity with conservation for nine-point scheme of nonlinear diffusion equations[J]. Comput Methods Appl Mech Eng, 2012, 223-224(1):161-172.
[9] NORDBOTTEN J, AAVATSMARK I, EIGESTAD G. Monotonicity of control volume methods[J]. Numer Math, 2007, 106(2):255-288.
[10] LE POTIER C. Finite volume monotone scheme for highly anisotropic diffusion operators on unstructured triangular meshes[J]. C R Math Acad Sci Paris, 2005, 341(12):787-792.
[11] LIPNIKOV K, SHASHKOV M, SVYATSKIY D, et al. Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes[J]. J Comput Phys, 2007, 227(1):492-512.
[12] LIPNIKOV K, SVYATSKIY D, VASSILEVSKI Y. Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes[J]. J Comput Phys, 2009, 228(3):703-716.
[13] LIPNIKOV K, SVYATSKIY D, VASSILEVSKI Y. A monotone finite volume method for advection-diffusion equations on unstructured polygonal meshes[J]. J Comput Phys, 2010, 229(11):4017-4032.
[14] YUAN G W, SHENG Z Q. Monotone finite volume schemes for diffusion equations on polygonal meshes[J]. J Comput Phys, 2008, 227(12):6288-6312.
[15] SHENG Z Q, YUE J Y, YUAN G W. Monotone finite volume schemes of nonequilibrium radiation diffusion equations on distorted meshes[J]. SIAM J Sci Comput, 2009, 31(4):2915-2934.
[16] SHENG Z Q, YUAN G W. An improved monotone finite volume scheme for diffusion equation on polygonal meshes[J]. J Comput Phys, 2012, 231(9):3739-3754.
[17] SHENG Z Q, YUAN G W. A new nonlinear finite volume scheme preserving positivity for diffusion equations[J]. J Comput Phys, 2016, 315:182-193.
[18] DANILOV A, VASSILEVSKI Y. A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes[J]. Russ J Numer Anal Math Modelling, 2009, 24(3):207-227.
[19] NIKITIN K, VASSILEVSKI Y. A monotone nonlinear finite volume method for advection-diffusion equations on unstructured polyhedral meshes in 3D[J]. Russ J Numer Anal Math Modelling, 2010, 25(4):335-358.
[20] HERMELINE F. Approximation of 2-D and 3-D diffusion operators with variable full tensor coefficients on arbitrary meshes[J]. Comput Methods Appl Mech Eng, 2007, 196(21):2497-2526.
[21] HERMELINE F. A finite volume method for approximating 3D diffusion operators on general meshes[J]. J Comput Phys, 2009, 228(16):5763-5786.
[22] LAI X, SHENG Z Q, YUAN G W. A finite volume scheme for three-dimensional diffusion equations[J]. Commun Comput Phys, 2015, 18(3):650-672.
[23] LAI X, SHENG Z Q, YUAN G W. Monotone finite volume scheme for three dimensional diffusion equation on tetrahedral meshes[J]. Commun Comput Phys, 2017, 21(1):162-181.
[24] WANG S, HANG X D, YUAN G W. A positivity-preserving finite volume scheme for diffusion equations on polyhedral meshes[J]. Math Numer Sin, 2015, 37(3):247-263.
[25] WANG S, HANG X D, YUAN G W. A pyramid scheme for three-dimensional diffusion equations on polyhedral meshes[J]. J Comput Phys, 2017, 350:590-606.
[26] WANG S, HANG X D, YUAN G W. A positivity-preserving pyramid scheme for anisotropic diffusion problems on general hexahedral meshes with nonplanar cell faces[J]. J Comput Phys, 2018, 371:152-167.
[27] LAN B, SHENG Z Q, YUAN G W. A new finite volume scheme preserving positivity for radionuclide transport calculations in radioactive waste repository[J]. Int J Heat Mass Tran, 2018, 121:736-746.
[28] GAO Y N, YUAN G W, WANG S, et al. A finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes[J]. J Comput Phys, 2020, 407:109-143.
[29] 袁光伟, 等. 扩散方程计算方法[M]. 北京:科学出版社, 2015.