A strongly-consistent difference scheme for 3D nonlinear Navier-Stokes equations. (English) Zbl 1483.65144

Summary: This paper constructs a strongly-consistent explicit finite difference scheme for 3D constant viscosity incompressible Navier-Stokes equations by using of symbolic algebraic computation. The difference scheme is space second order accurate and temporal first order accurate. It is proved that difference Gröbner basis algorithm is correct. By using of difference Gröbner basis computation method, an element in Gröbner basis of difference scheme for momentum equations is a difference scheme for pressure Poisson equation. The authors find that the truncation errors expressions of difference scheme is consistent with continuous errors functions about modified version of above difference equation. The authors prove that, for strongly consistent difference scheme, each element in the difference Gröbner basis of such difference scheme always approximates a differential equation which vanishes on the analytic solutions of Navier-Stokes equations. To prove the strongly-consistency of this difference scheme, the differential Thomas decomposition theorem for nonlinear differential equations and difference Gröbner basis theorems for difference equations are applied. Numerical test certifies that strongly-consistent difference scheme is effective.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q30 Navier-Stokes equations
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
68W30 Symbolic computation and algebraic computation


Full Text: DOI


[1] Anderson, J. D., Computational fuild dynamics: The basics with applications, McGraw-Hill Series in Mechanical Engineering (1995), New York: McGraw-Hill College McGraw-Hill Science, New York
[2] Levin, A., Difference Algebra, Algebra and Applications (2008), New York: Springer, New York · Zbl 1209.12003
[3] Amodio, P.; Blinkov, Y. A.; Gerdt, V. P., On consistency of finite difference approximations to the Navier-Stokes equations, Computer Algebra in Scientific Computing, 46-60 (2013), Cham, Switzerland: Springer, Cham, Switzerland · Zbl 1412.65063
[4] Amodio, P.; Blinkov, Y. A.; Gerdt, V. P., Algebraic construction and numerical behavior of a new s-consistent difference scheme for the 2D Navier-stokes equations, Applied Mathematics and Computation, 314, 408-421 (2017) · Zbl 1426.76449
[5] Blinkov, Y. A.; Gerdt, V. P.; Lyakhov, D. A., A strongly consistent finite difference scheme for steady stokes flow and its modified equations, Computer Algebra in Scientific Computing, 67-81 (2018), Cham, Switzerland: Springer, Cham, Switzerland · Zbl 1453.76122
[6] Zhang, X. J.; Gerdt, V. P.; Blinkov, Y. A., Algebraic construction of a strongly consistent, permutationally symmetric and conservative difference scheme for 3D steady stokes flow, Symmetry, 11, 2, 269-283 (2019) · Zbl 1416.76196
[7] Gerdt V P, Involutive algorithms for computing Gröbner bases, Mathematics, 2005, 174-195. · Zbl 1104.13012
[8] Gerdt, V. P.; Blinkov, Y. A., Involutive bases of polynomial ideals, Mathematics and Computers in Simulation, 45, 519-541 (1998) · Zbl 1017.13500
[9] Gerdt, V. P.; Robertz, D., A Maple package for computing Gröbner bases for linear recurrence relations, Nuclear Inst & Methods in Physics Research A, 559, 1, 215-219 (2005)
[10] LaScala, R., Gröbner bases and grading for partial difference ideals, Mathematics of Computation, 84, 292, 959-985 (2011) · Zbl 1328.12014
[11] Gerdt, V. P.; LaScala, R., Noetherian quotients of the algebra of partial difference polynomials and Gröbner bases of symmetric ideals, Journal of Algebra, 423, 1233-1261 (2015) · Zbl 1327.12003
[12] Robertz, D., Formal algorithmic elimination for PDEs, Lecture Notes in Mathematics (2014), Cham: Springer, Cham · Zbl 1339.35007
[13] Gerdt, V. P.; Lange-Hegermann, M.; Robertz, D., The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs, Computer Physics Communications, 234, 202-215 (2019)
[14] Chen, Y. F.; Gao, X. S., Involutive characteristic sets of algebraic partial differential equation systems, Science in China, 46, 4, 469-487 (2003) · Zbl 1215.35018
[15] Chen, Y. F.; Gao, X. S., Involutive bases of algebraic partial differential equation systems, MM Research Preprints, Beijing, 19, 11-32 (2000)
[16] Gerdt, V. P., Completing of linear differential systems to involution, Computer Algebra in Scientific Computing CASC’99 (1999), Verlag Berlin Heidelberg: Springer, Verlag Berlin Heidelberg
[17] Chen, Y. F., Lectures on Computer Algebra (2009), Beijing: Higher Education Press, Beijing
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.