2013, 2013(special): 489-497. doi: 10.3934/proc.2013.2013.489

A discontinuous Galerkin least-squares finite element method for solving Fisher's equation

1. 

Department of Engineering, Mathematics, and Physics, Texas A&M International University, Laredo, TX 78041

2. 

Department of Mathematics, The University of Southern Mississippi, Hattiesburg, MS 39406

Received  September 2012 Revised  January 2013 Published  November 2013

In the present study, a discontinuous Galerkin least-squares finite element algorithm is developed to solve Fisher's equation. The present method is effective and can be successfully applied to problems with strong reaction, to which obtaining stable and accurate numerical traveling wave solutions is challenging. Numerical results are given to demonstrate the convergence rates of the method and the performance of the algorithm in long-time integrations.
Citation: Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489
References:
[1]

M.J. Ablowitz and A. Zeppetella, Explicit solutions of Fisher's equation for a special wave speed, Bull. Math. Biol., 41 (1979), no. 6, pp. 835-840.  Google Scholar

[2]

K. Al-Khaled, Numerical study of Fishers reaction-diffusion equation by the sinc collocation method, J. Comput. Appl. Math., 137 (2001), pp. 245-255.  Google Scholar

[3]

J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Develop., 17 (1973), pp. 307-313.  Google Scholar

[4]

G.F. Carey and Y. Shen, Least-squares finite element approximation of Fishers reactiondiffusion equation, Numer. Methods Partial Differential Equations, 11 (1995), pp. 175-186.  Google Scholar

[5]

I. Daǧ, A. Şahin, and A. Korkmaz, Numerical investigation of the solution of Fisher's equation via the B-spline Galerkin method, Numer. Methods Partial Differential Equations 26 (2010), no. 6, pp. 1483-1503.  Google Scholar

[6]

R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), pp. 355-369. Google Scholar

[7]

J. Gazdag and J. Canosa, Numerical solution of Fisher's equation, J. Appl. Probab., 11 (1974), pp. 445-457.  Google Scholar

[8]

B.Y. Guo and Z.X. Chen, Analytic solutions of the Fisher equation, J. Phys. A, 24 (1991), no. 3, pp. 645-650.  Google Scholar

[9]

P.S. Hagan, Traveling wave and multiple traveling wave solutions of parabolic equations, SIAM J. Math. Anal. 13 (1982), no. 5, pp. 717-738.  Google Scholar

[10]

T. Hagstrom and H.B. Keller, The numerical calculation of traveling wave solutions of nonlinear parabolic equations, SIAM J. Sci. Statist. Comput., 7 (1986), no. 3, pp. 978-988.  Google Scholar

[11]

A. Kolmogorov, I. Petrovshy, and N. Piscounoff, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. Etat Moscou Ser. Int. Sect. A Math. et Mecan., 1 (1937), pp. 1-25.  Google Scholar

[12]

D.A. Larson, Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type, SIAM J. Appl. Math. 34 (1978), no. 1, pp. 93-103.  Google Scholar

[13]

S. Li, L. Petzold, and Y. Ren, Stability of moving mesh systems of partial differential equations, SIAM J. Sci. Comput., 20 (1998), no. 2, pp. 719-738.  Google Scholar

[14]

R. Lin, Discontinuous discretization for least-squares formulation of singularly perturbed reaction-diffusion problems in one and two dimensions,, SIAM J. Numer. Anal. 47 (2008/09), 47 (): 89.   Google Scholar

[15]

R. Lin, Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities, Numer. Math. 112 (2009), no. 2, pp. 295-318.  Google Scholar

[16]

J.D. Logan, "An introduction to nonlinear partial differential equations,'' second edition, Wiley-Interscience, John Wiley & Sons, Hoboken, NJ, 2008.  Google Scholar

[17]

R.E. Mickens, A best finite-difference scheme for the Fisher equation, Numer. Methods Partial Differential Equations 10 (1994), no. 5, pp. 581-585.  Google Scholar

[18]

J.D. Murray, "Mathematical biology,'' Biomathematics, 19, Springer-Verlag, Berlin, 1989.  Google Scholar

[19]

D. Olmos and B.D. Shizgal, A pseudospectral method of solution of Fisher's equation, J. Comput. Appl. Math., 193 (2006), pp. 219-242.  Google Scholar

[20]

N. Parekh and S. Puri, A new numerical scheme for the Fisher equation, J. Phys. A: Math. Gen., 23 (1990), pp. L1085-L1091.  Google Scholar

[21]

Y. Qiu and D.M. Sloan, Numerical solution of Fisher's equation using a moving mesh method, J. Comput. Phys., 146 (1998), pp. 726-746.  Google Scholar

[22]

Rizwan-uddin, Comparison of the nodal integral method and nonstandard finite-difference schemes for the Fisher equation, SIAM. J. Sci. Comput., 22 (2000), pp. 1926-1942.  Google Scholar

[23]

J. Roessler and H. Hüssner, Numerical solution of the $1+2$ dimensional Fisher's equation by finite elements and the Galerkin method, Math. Comput. Modelling, 25 (1997), pp. 57-67.  Google Scholar

[24]

S. Tang and R.O. Weber, Numerical study of Fisher's equation by a Petrov-Galerkin finite element method, J. Austral. Math. Soc. Sci. B, 33 (1991) pp. 27-38.  Google Scholar

[25]

V. Thomée, "Galerkin finite element methods for parabolic problems,'' second edition, Springer Series in Computational Mathematics, 25, Springer-Verlag, Berlin, 2006.  Google Scholar

[26]

S. Zhao and G.W. Wei, Comparison of the discrete singular convolution and three other numerical schemes for solving Fisher's equation, SIAM J. Sci. Comput., 25 (2003) pp. 127-147.  Google Scholar

show all references

References:
[1]

M.J. Ablowitz and A. Zeppetella, Explicit solutions of Fisher's equation for a special wave speed, Bull. Math. Biol., 41 (1979), no. 6, pp. 835-840.  Google Scholar

[2]

K. Al-Khaled, Numerical study of Fishers reaction-diffusion equation by the sinc collocation method, J. Comput. Appl. Math., 137 (2001), pp. 245-255.  Google Scholar

[3]

J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Develop., 17 (1973), pp. 307-313.  Google Scholar

[4]

G.F. Carey and Y. Shen, Least-squares finite element approximation of Fishers reactiondiffusion equation, Numer. Methods Partial Differential Equations, 11 (1995), pp. 175-186.  Google Scholar

[5]

I. Daǧ, A. Şahin, and A. Korkmaz, Numerical investigation of the solution of Fisher's equation via the B-spline Galerkin method, Numer. Methods Partial Differential Equations 26 (2010), no. 6, pp. 1483-1503.  Google Scholar

[6]

R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), pp. 355-369. Google Scholar

[7]

J. Gazdag and J. Canosa, Numerical solution of Fisher's equation, J. Appl. Probab., 11 (1974), pp. 445-457.  Google Scholar

[8]

B.Y. Guo and Z.X. Chen, Analytic solutions of the Fisher equation, J. Phys. A, 24 (1991), no. 3, pp. 645-650.  Google Scholar

[9]

P.S. Hagan, Traveling wave and multiple traveling wave solutions of parabolic equations, SIAM J. Math. Anal. 13 (1982), no. 5, pp. 717-738.  Google Scholar

[10]

T. Hagstrom and H.B. Keller, The numerical calculation of traveling wave solutions of nonlinear parabolic equations, SIAM J. Sci. Statist. Comput., 7 (1986), no. 3, pp. 978-988.  Google Scholar

[11]

A. Kolmogorov, I. Petrovshy, and N. Piscounoff, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. Etat Moscou Ser. Int. Sect. A Math. et Mecan., 1 (1937), pp. 1-25.  Google Scholar

[12]

D.A. Larson, Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type, SIAM J. Appl. Math. 34 (1978), no. 1, pp. 93-103.  Google Scholar

[13]

S. Li, L. Petzold, and Y. Ren, Stability of moving mesh systems of partial differential equations, SIAM J. Sci. Comput., 20 (1998), no. 2, pp. 719-738.  Google Scholar

[14]

R. Lin, Discontinuous discretization for least-squares formulation of singularly perturbed reaction-diffusion problems in one and two dimensions,, SIAM J. Numer. Anal. 47 (2008/09), 47 (): 89.   Google Scholar

[15]

R. Lin, Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities, Numer. Math. 112 (2009), no. 2, pp. 295-318.  Google Scholar

[16]

J.D. Logan, "An introduction to nonlinear partial differential equations,'' second edition, Wiley-Interscience, John Wiley & Sons, Hoboken, NJ, 2008.  Google Scholar

[17]

R.E. Mickens, A best finite-difference scheme for the Fisher equation, Numer. Methods Partial Differential Equations 10 (1994), no. 5, pp. 581-585.  Google Scholar

[18]

J.D. Murray, "Mathematical biology,'' Biomathematics, 19, Springer-Verlag, Berlin, 1989.  Google Scholar

[19]

D. Olmos and B.D. Shizgal, A pseudospectral method of solution of Fisher's equation, J. Comput. Appl. Math., 193 (2006), pp. 219-242.  Google Scholar

[20]

N. Parekh and S. Puri, A new numerical scheme for the Fisher equation, J. Phys. A: Math. Gen., 23 (1990), pp. L1085-L1091.  Google Scholar

[21]

Y. Qiu and D.M. Sloan, Numerical solution of Fisher's equation using a moving mesh method, J. Comput. Phys., 146 (1998), pp. 726-746.  Google Scholar

[22]

Rizwan-uddin, Comparison of the nodal integral method and nonstandard finite-difference schemes for the Fisher equation, SIAM. J. Sci. Comput., 22 (2000), pp. 1926-1942.  Google Scholar

[23]

J. Roessler and H. Hüssner, Numerical solution of the $1+2$ dimensional Fisher's equation by finite elements and the Galerkin method, Math. Comput. Modelling, 25 (1997), pp. 57-67.  Google Scholar

[24]

S. Tang and R.O. Weber, Numerical study of Fisher's equation by a Petrov-Galerkin finite element method, J. Austral. Math. Soc. Sci. B, 33 (1991) pp. 27-38.  Google Scholar

[25]

V. Thomée, "Galerkin finite element methods for parabolic problems,'' second edition, Springer Series in Computational Mathematics, 25, Springer-Verlag, Berlin, 2006.  Google Scholar

[26]

S. Zhao and G.W. Wei, Comparison of the discrete singular convolution and three other numerical schemes for solving Fisher's equation, SIAM J. Sci. Comput., 25 (2003) pp. 127-147.  Google Scholar

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