Mimetic methods to Helmholtz equation: numerical dispersion
Keywords:
numerical dispersion, mimetic methods, acoustic scattering, Helmholtz equation
Abstract
This is the rst in a series of papers in where mimetic nite dierence methods (MFDM) to acoustic scattering are applying. A simple one-dimensional problem has been chosen to illustrate an implementation of MFDM. This problem consists of an incident at pressure wave that disperses from an innite rigid wall. An absorbent boundary condition is also applied. A study of the order of convergence and numerical dispersion of these methods has been carried out.
References
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M. Ainsworth and H. A. Wajid. Dispersive and dissipative behavior of the spectral element method. SIAM Journal on Numerical Analysis. 47 (2009) 3910-3937.
I. M. Babuška and S. A. Sauter. Is the pollution effect of the fem avoidable for the helmholtz equation considering high wave numbers?. SIAM Journal on numerical analysis. 34 (1997) 2392-2423.
M. Bartoň, V. Calo, Q. Deng, and V. Puzyrev. Generalization of the Pythagorean Eigenvalue Error Theorem and its Application to Isogeometric Analysis. In Di Pietro D., Ern A., Formaggia L. (eds) Numerical Methods for PDEs. SEMA SIMAI Springer Series, vol 15. Springer, Cham. (2018) 147-170.
L. Beirão da Veiga, V. Gyrya, K. Lipnikov and G. Manzini. Mimetic finite difference method for the Stokes problem on polygonal meshes. Journal of Computational Physics. 228 (2009) 7215-7232.
J. Blanco, O. Rojas, C. Chacón, J. M. Guevara-Jordan and J. Castillo. Tensor formulation of 3-D mimetic finite differences and applications to elliptic problems. Electronic Transactions on Numerical Analysis. 45 (2016) 457-475.
C. E. Cadenas R. Formulación y Aplicación del Método de Elementos Finitos Mı́nimos Cuadrados a un Problema de Dispersión de Ondas y Comparación con otros Métodos Numéricos. Trabajo de Ascenso, Universidad de Carabobo. (2003). pp. 115.
C. E. Cadenas R. and V. Villamizar Application of Least Squares Finite Element Method to Acoustic Scattering and Comparison with Other Numerical Techniques. NACoM-2003, Cambridge. Extended Abstracts. (2003) 32-35.
C. E. Cadenas R. and V. Villamizar. Comparison of Least Squares FEM, Mixed Galerkin FEM and an Implicit FDM Applied to Acoustic Scattering. Appl. Num. Anal. Comp. Math. 1 (1) (2004) 128-139.
C. E. Cadenas R. and O. Montilla M. Matriz de Vandermonde generalizada para la construcción de los operadores de divergencia discreta mimética. Revista Ingenierı́a UC. 11 (2) (2004) 48-52.
C. E. Cadenas, J. J. Rojas and V. Villamizar. A least squares finite element method with high degree element shape functions for one-dimensional Helmholtz equation. Mathematics and Computers in Simulation. 73 (2006) 76-86
C. E. Cadenas, y L. J. Quiñonez. Resolviendo la ecuación de Helmholtz 1D por métodos miméticos. 1er Congreso Venezolano de Ciencia Tecnologı́a e Innovación, LOCTI-PEII. Caracas, 23 - 26 de septiembre de 2012, Caracas, Venezuela.
J. E. Castillo and M. Yasuda. Comparison of two matrix operator formulations for mimetic divergence and gradient discretizations. International Conference on Parallel and Distributed Processing Techniques and Applications. Volume: III. (2003)
J. E. Castillo and R. D. Grone. A Matrix Analysis Approach to Higher-Order Approximations for Divergence and Gradients Satisfying a Global Conservation Law, SIAM J MATRIX ANAL, 25 (1) (2003) 128-142 .
J. E. Castillo and G. Miranda. Mimetic discretization methods. CRC Press, Taylor and Francis Group (2013), pp 230.
J. E. Castillo and R. D. Grone. Using Kronecker products to construct mimetic gradients. Linear and Multilinear Algebra. 65 (10) (2017) 2031-2045.
J. E. Castillo and G. Miranda. High Order Compact Mimetic Differences and Discrete Energy Decay in 2D Wave Motions. In: Bittencourt M., Dumont N., Hesthaven J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Lecture Notes in Computational Science and Engineering, vol 119. (2017). Springer, Cham.
M, Colmenares. Métodos Miméticos Aplicados a la Ecuación de Helmholtz 1D Sobre Mallados No-Uniformes, Trabajo Especial de Grado para optar al Tı́tulo de Licenciada en Matemática. Facultad Experimental de Ciencias y Tecnologá de la Universidad de Carabobo. Valencia, 2008.
J. Corbino and J. Castillo. MOLE: Mimetic Operators Library Enhanced The Open-Source Library for Solving Partial Differential Equations using Mimetic Methods. Computational Science and Engineering Faculty and Students Research Articles. Technical Report. (2017).
M. Fagúndez, J. Medina, C. Cadenas and G. Larrazabal. Discretizaciones Miméticas para dinámica de fluidos computacional: Caso Unidimensional. Revista Ingenierı́a UC. 11(3) (2004), 52-57.
V. Gyrya and K. Lipnikov. High-order mimetic finite difference method for diffusion problems on polygonal meshes. Journal of Computational Physics. 227 (2008), 8841-8854.
V. Gyrya and K. Lipnikov. The arbitrary order mimetic finite difference method for a diffusion equation with a non-symmetric diffusion tensor. Journal of Computational Physics. 348(1) (2017), 549-566.
I. Harari. Reducing spurious dispersion, anisotropy and reflection in finite element analysis of time-harmonic acoustics. Computer methods in applied mechanics and engineering. 140 (1997), 39-58.
F. Ihlenburg and I. Babuška, Dispersion analysis and error of Galerkin finite element methods for the Helmholtz equation. International journal for numerical methods in engineering. 38 (1995), 3745-3774.
H. Levy and F. Lessman. Finite Difference Equations. Dover. New York, pp. 278. (1992).
K. Lipnikov, G. Manzini and D. Svyatskiy. Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems. Journal of Computational Physics. 230 (2011), 2620-2642.
K. Lipnikov, G. Manzini, F. Brezzi and A. Buffa. The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes. Journal of Computational Physics. 230 (2011), 305-328.
K. Lipnikov and G. Manzini. A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation. Journal of Computational Physics. 272 (2014), 360-385.
K. Lipnikov, G. Manzini and M. Shashkov. Mimetic finite difference method. Journal of Computational Physics. 257 (2014), 1163-1227.
O. Montilla, C. Cadenas and J. Castillo. Matrix Approach to Mimetic Discretizations for Differential Operators on Non-uniform Grids. Mathematics and Computers in Simulation. 73 (2006), 215-225.
G. T. Oud, D. R. van der Heul, C. Vuik and R.A.W.M. Henkes. A fully conservative mimetic discretization of the NavierStokes equations in cylindrical coordinates with associated singularity treatment. Journal of Computational Physics. 325(15) (2016), 314-337.
S. Rojas. Mimetic Finite Difference Method for the Steady Diffusion Equation with Rough Coefficients. 2003 SIAM Conference on Computational Science and Engineering, Mimetic Computations, San Diego, CA, USA. (2003).
A. A. Samarskii, V. F. Tishkin, A. P. Favorskii and M. Shashkov Operational Finite-Difference Schemes, Diff. Eqns., 17 No. 7, 854-862, (1981).
A. A. Samarskii, V. F. Tishkin, A. P. Favorskii, and M. Shashkov. Employment of the Reference-Operator Method in the Construction of Finite Difference Analogs of Tensor Operations, Diff. Eqns., 18, No. 7, 881-885, (1982).
E. Sanchez, C. Paolini, P. Blomgren and J. Castillo. Algorithms for Higher-Order Mimetic Operators. In: Kirby R., Berzins M., Hesthaven J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol 106. (2015). Springer, Cham.
E. J. Sanchez, G. F. Miranda, J. M. Cela, and J. E. Castillo. Supercritical-Order Mimetic Operators on Higher-Dimensional Staggered Grids. In: Bittencourt M., Dumont N., Hesthaven J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Lecture Notes in Computational Science and Engineering, vol 119. (2017). Springer, Cham.
M. Shashkov. Conservative Finite-Difference Methods on General Grids. CRC Press. Florida, USA (1996).
G. Sosa, J. Arteaga and O. Jiménez. A study of mimetic and finite difference methods for the static diffusion equation. Computers and Mathematics with Applications. 76(3) (2018), 633-648.
M. Ainsworth and H. A. Wajid. Dispersive and dissipative behavior of the spectral element method. SIAM Journal on Numerical Analysis. 47 (2009) 3910-3937.
I. M. Babuška and S. A. Sauter. Is the pollution effect of the fem avoidable for the helmholtz equation considering high wave numbers?. SIAM Journal on numerical analysis. 34 (1997) 2392-2423.
M. Bartoň, V. Calo, Q. Deng, and V. Puzyrev. Generalization of the Pythagorean Eigenvalue Error Theorem and its Application to Isogeometric Analysis. In Di Pietro D., Ern A., Formaggia L. (eds) Numerical Methods for PDEs. SEMA SIMAI Springer Series, vol 15. Springer, Cham. (2018) 147-170.
L. Beirão da Veiga, V. Gyrya, K. Lipnikov and G. Manzini. Mimetic finite difference method for the Stokes problem on polygonal meshes. Journal of Computational Physics. 228 (2009) 7215-7232.
J. Blanco, O. Rojas, C. Chacón, J. M. Guevara-Jordan and J. Castillo. Tensor formulation of 3-D mimetic finite differences and applications to elliptic problems. Electronic Transactions on Numerical Analysis. 45 (2016) 457-475.
C. E. Cadenas R. Formulación y Aplicación del Método de Elementos Finitos Mı́nimos Cuadrados a un Problema de Dispersión de Ondas y Comparación con otros Métodos Numéricos. Trabajo de Ascenso, Universidad de Carabobo. (2003). pp. 115.
C. E. Cadenas R. and V. Villamizar Application of Least Squares Finite Element Method to Acoustic Scattering and Comparison with Other Numerical Techniques. NACoM-2003, Cambridge. Extended Abstracts. (2003) 32-35.
C. E. Cadenas R. and V. Villamizar. Comparison of Least Squares FEM, Mixed Galerkin FEM and an Implicit FDM Applied to Acoustic Scattering. Appl. Num. Anal. Comp. Math. 1 (1) (2004) 128-139.
C. E. Cadenas R. and O. Montilla M. Matriz de Vandermonde generalizada para la construcción de los operadores de divergencia discreta mimética. Revista Ingenierı́a UC. 11 (2) (2004) 48-52.
C. E. Cadenas, J. J. Rojas and V. Villamizar. A least squares finite element method with high degree element shape functions for one-dimensional Helmholtz equation. Mathematics and Computers in Simulation. 73 (2006) 76-86
C. E. Cadenas, y L. J. Quiñonez. Resolviendo la ecuación de Helmholtz 1D por métodos miméticos. 1er Congreso Venezolano de Ciencia Tecnologı́a e Innovación, LOCTI-PEII. Caracas, 23 - 26 de septiembre de 2012, Caracas, Venezuela.
J. E. Castillo and M. Yasuda. Comparison of two matrix operator formulations for mimetic divergence and gradient discretizations. International Conference on Parallel and Distributed Processing Techniques and Applications. Volume: III. (2003)
J. E. Castillo and R. D. Grone. A Matrix Analysis Approach to Higher-Order Approximations for Divergence and Gradients Satisfying a Global Conservation Law, SIAM J MATRIX ANAL, 25 (1) (2003) 128-142 .
J. E. Castillo and G. Miranda. Mimetic discretization methods. CRC Press, Taylor and Francis Group (2013), pp 230.
J. E. Castillo and R. D. Grone. Using Kronecker products to construct mimetic gradients. Linear and Multilinear Algebra. 65 (10) (2017) 2031-2045.
J. E. Castillo and G. Miranda. High Order Compact Mimetic Differences and Discrete Energy Decay in 2D Wave Motions. In: Bittencourt M., Dumont N., Hesthaven J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Lecture Notes in Computational Science and Engineering, vol 119. (2017). Springer, Cham.
M, Colmenares. Métodos Miméticos Aplicados a la Ecuación de Helmholtz 1D Sobre Mallados No-Uniformes, Trabajo Especial de Grado para optar al Tı́tulo de Licenciada en Matemática. Facultad Experimental de Ciencias y Tecnologá de la Universidad de Carabobo. Valencia, 2008.
J. Corbino and J. Castillo. MOLE: Mimetic Operators Library Enhanced The Open-Source Library for Solving Partial Differential Equations using Mimetic Methods. Computational Science and Engineering Faculty and Students Research Articles. Technical Report. (2017).
M. Fagúndez, J. Medina, C. Cadenas and G. Larrazabal. Discretizaciones Miméticas para dinámica de fluidos computacional: Caso Unidimensional. Revista Ingenierı́a UC. 11(3) (2004), 52-57.
V. Gyrya and K. Lipnikov. High-order mimetic finite difference method for diffusion problems on polygonal meshes. Journal of Computational Physics. 227 (2008), 8841-8854.
V. Gyrya and K. Lipnikov. The arbitrary order mimetic finite difference method for a diffusion equation with a non-symmetric diffusion tensor. Journal of Computational Physics. 348(1) (2017), 549-566.
I. Harari. Reducing spurious dispersion, anisotropy and reflection in finite element analysis of time-harmonic acoustics. Computer methods in applied mechanics and engineering. 140 (1997), 39-58.
F. Ihlenburg and I. Babuška, Dispersion analysis and error of Galerkin finite element methods for the Helmholtz equation. International journal for numerical methods in engineering. 38 (1995), 3745-3774.
H. Levy and F. Lessman. Finite Difference Equations. Dover. New York, pp. 278. (1992).
K. Lipnikov, G. Manzini and D. Svyatskiy. Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems. Journal of Computational Physics. 230 (2011), 2620-2642.
K. Lipnikov, G. Manzini, F. Brezzi and A. Buffa. The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes. Journal of Computational Physics. 230 (2011), 305-328.
K. Lipnikov and G. Manzini. A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation. Journal of Computational Physics. 272 (2014), 360-385.
K. Lipnikov, G. Manzini and M. Shashkov. Mimetic finite difference method. Journal of Computational Physics. 257 (2014), 1163-1227.
O. Montilla, C. Cadenas and J. Castillo. Matrix Approach to Mimetic Discretizations for Differential Operators on Non-uniform Grids. Mathematics and Computers in Simulation. 73 (2006), 215-225.
G. T. Oud, D. R. van der Heul, C. Vuik and R.A.W.M. Henkes. A fully conservative mimetic discretization of the NavierStokes equations in cylindrical coordinates with associated singularity treatment. Journal of Computational Physics. 325(15) (2016), 314-337.
S. Rojas. Mimetic Finite Difference Method for the Steady Diffusion Equation with Rough Coefficients. 2003 SIAM Conference on Computational Science and Engineering, Mimetic Computations, San Diego, CA, USA. (2003).
A. A. Samarskii, V. F. Tishkin, A. P. Favorskii and M. Shashkov Operational Finite-Difference Schemes, Diff. Eqns., 17 No. 7, 854-862, (1981).
A. A. Samarskii, V. F. Tishkin, A. P. Favorskii, and M. Shashkov. Employment of the Reference-Operator Method in the Construction of Finite Difference Analogs of Tensor Operations, Diff. Eqns., 18, No. 7, 881-885, (1982).
E. Sanchez, C. Paolini, P. Blomgren and J. Castillo. Algorithms for Higher-Order Mimetic Operators. In: Kirby R., Berzins M., Hesthaven J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol 106. (2015). Springer, Cham.
E. J. Sanchez, G. F. Miranda, J. M. Cela, and J. E. Castillo. Supercritical-Order Mimetic Operators on Higher-Dimensional Staggered Grids. In: Bittencourt M., Dumont N., Hesthaven J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Lecture Notes in Computational Science and Engineering, vol 119. (2017). Springer, Cham.
M. Shashkov. Conservative Finite-Difference Methods on General Grids. CRC Press. Florida, USA (1996).
G. Sosa, J. Arteaga and O. Jiménez. A study of mimetic and finite difference methods for the static diffusion equation. Computers and Mathematics with Applications. 76(3) (2018), 633-648.
Published
2019-06-29
How to Cite
Cadenas R., C. E., & Quiñonez T., L. J. (2019). Mimetic methods to Helmholtz equation: numerical dispersion. Divulgaciones Matemáticas, 20(1), 1-15. Retrieved from https://produccioncientificaluz.org/index.php/divulgaciones/article/view/36618
Section
Research papers
