Stability of explicit schemes for solving Maxwell’s equations by high-order finite volume methods


  • D.K. Firsov Geomodeling Technology Corp.


Maxwell’s equations, finite volume method, stability of explicit schemes, high-order accuracy, partial differential equations


A new stability criterion of explicit schemes for solving Maxwell’s equations by high-order finite volume methods is proposed. The proof is based on a generalization of the stability criterion for the first-order finite volume scheme to the case of high-order schemes. The effect of discontinuities of the solution on the stability of high-order schemes is evaluated. The maximum principle for the finite volume approximations of vector conservation laws is discussed.

Author Biography

D.K. Firsov

Geomodeling Technology Corp.
• Numerical Programmer


  1. Fumeaux C., Baumann D., Leuchtmann P., Vahldieck R. A generalized local time-step scheme for efficient FVTD simulations in strongly inhomogeneous meshes // IEEE Transactions on Microwave Theory and Techniques. 2004. 52, N 3. 1067-1076.
  2. Lauritzen P.H. A stability analysis of finite-volume advection schemes permitting long time steps // Monthly Weather Reviews. 2007. 135, N 7. 2658-2673.
  3. Piperno S. L_2-stability of the upwind first order finite volume scheme for the Maxwell equations in two and three dimensions on arbitrary unstructured meshes // Mathematical Modelling and Numerical Analysis. 2000. 34, N 1. 139-158.
  4. Piperno S. Symplectic local time-stepping in non-dissipative DGTG methods applied to wave propagation problems // Mathematical Modelling and Numerical Analysis. 2006. 40, N 5. 815-841.
  5. Calgaro C., Chane-Kane E., Creusé E., Goudon T. L_infty -stability of vertex-based MUSCL finite volume schemes on unstructured grids: simulation of incompressible flows with high density ratios // Journal of Computational Physics. 2010. 229, N 17. 6027-6046.
  6. Coudiére Y., Pierre C. Stability and convergence of a finite volume method for two systems of reaction-diffusion equations in electro-cardiology // Nonlinear Analysis: Real World Applications. 2006. 7, N 4. 916-935.
  7. Шурина Э.П., Великая М.Ю., Федорук М.П. Об алгоритмах решения уравнений Максвелла на неструктурированных сетках // Вычислительные технологии. 2000. 5, № 6. 99-116.
  8. Firsov D., LoVetri J. New stability criterion for unstructured mesh upwinding FVTD schemes for Maxwell’s equations // ACES Journal. 2008. 23, N 3. 193-199.
  9. Gottlieb S., Shu C.-W., Tadmor E. Strong stability-preserving high-order time discretization methods // SIAM Review. 2001. 43, N 1. 89-112.
  10. Лебедев А.С., Федорук М.П., Штырина О.В. Решение нестационарных уравнений Максвелла для сред с неоднородными свойствами методом конечных объемов // Вычислительные технологии. 2005. 10, № 2. 60-73.
  11. Bonnet P., Ferrieres X., Michielsen B.L., Klotz P. Finite-volume time domain method // Time Domain Electromagnetics. New York: Academic Press, 1999. 307-367.
  12. Chung E.T., Engquist B. Convergence analysis of fully discrete finite volume methods for Maxwell’s equations in nonhomogeneous media // SIAM Journal on Numerical Analysis. 2006. 43, N 1. 303-317.
  13. Barth T., Ohlberger M. Finite volume methods: foundation and analysis // Encyclopedia of Computational Mechanics. Vol. 1. New York: Wiley, 2004. 1-57.
  14. Firsov D., LoVetri J., Jeffrey I., Okhmatovski V., Gilmore C., Chamma W. High-order FVTD on unstructured grids using an object-oriented computational engine // ACES Journal. 2007. 22, N 1. 71-82.
  15. Firsov D., LoVetri J. New stability criterion for unstructured mesh upwinding FVTD schemes for Maxwell’s equations // Proc. of the 23th Annual Review of Progress in Applied Computational Electromagnetics. Verona: ACES Press, 2007. 401-408.
  16. Hubbard M.E. Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids // Journal of Computational Physics. 1999. 155, N 1. 54-74.
  17. Clain S. Finite volume maximum principle for hyperbolic scalar problems // SIAM J. Numer. Anal. 2013. 51, N 1. 467-490.
  18. Harrington R.F. Time-harmonic electromagnetic fields. New York: McGraw-Hill, 1961.
  19. Kaye C., Gilmore C., Mojabi P., Firsov D., LoVetri J. Development of a resonant chamber microwave tomography system // Ultra-Wideband, Short Pulse Electromagnetics. Vol. 9. New York: Springer, 2010. 481-488.



How to Cite

Фирсов Д.К. Stability of Explicit Schemes for Solving Maxwell’s Equations by High-Order Finite Volume Methods // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2014. 15. 286-303



Section 1. Numerical methods and applications