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Hybrid radial basis function methods of lines for the numerical solution of viscous Burgers’ equation

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Abstract

An efficient and accurate hybrid radial basis function (HRBF) method is proposed to numerically solve the quasi-linear viscous Burgers’ equation. Two different RBFs are combined: an infinite smooth RBF defined by a shape parameter and a piecewise smooth RBF independent of the shape parameter. This combination formulates a family of HRBF methods, which is then used to approximate the spatial operator of the viscous Burgers equation. An efficient high-order solver is then used to integrate in time the resulting initial value problem. The developed numerical scheme is tested on some benchmark problems varying the Reynolds number. Accuracy and efficiency is validated using \(L_{\infty },~ L_{2}\) and \(L_{\text {rms}}\) error norms, as well as the number of nodes N over the domain of influence. A rigorous comparison is conducted against well-known numerical schemes to manifest improved accuracy of the proposed scheme. Eigenvalue stability analysis of the proposed technique is thoroughly discussed and confirmed by numerical examples. The proposed scheme is found a well-behaved alternative to RBF (Kansa) and RBF-PS methods with improved accuracy and good conditioning properties.

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References

  • Ahmed SG (2006) A collocation method using new combined radial basis functions of thin plate and multiquadraic types. Eng Anal Bound Elem 30(8):697–701

    MATH  Google Scholar 

  • Ali A, Mukhtar S, Hussain I (2011) A numerical meshless technique for the solution of some Burgers’ type equations. World Appl Sci J 14(12):1792–1798

    Google Scholar 

  • Asaithambi A (2010) Numerical solution of the Burgers’ equation by automatic differentiation. Appl Math Comput 216:2700–2708

    MathSciNet  MATH  Google Scholar 

  • Biancolini ME (2017) Fast radial basis functions for engineering applications. Springer International Publishing AG, Springer Nature, Berlin

    MATH  Google Scholar 

  • Brezis H, Browder F (1998) Partial differential equations in the 20th century. Adv Math 135:76–144

    MathSciNet  MATH  Google Scholar 

  • Burgers JM (1948) A mathematical model illustrating the theory of turbulence. Adv Appl Mech 1:171–199

    MathSciNet  Google Scholar 

  • Cavoretto R (2015) A numerical algorithm for multidimensional modeling of scattered data points. Comput Appl Math 34:65–80

    MathSciNet  MATH  Google Scholar 

  • Cavoretto R, De Rossi A, Mukhametzhanov MS et al (2019) On the search of the shape parameter in radial basis functions using univariate global optimization methods. J Glob Optim. https://doi.org/10.1007/s10898-019-00853-3

    Article  MATH  Google Scholar 

  • Cecil T, Qian J, Osher S (2004) Numerical methods for high dimensional Hamilton–Jacobi equations using radial basis functions. J Comput Phys 196(1):327–347

    MathSciNet  MATH  Google Scholar 

  • Chan YL, Shen LH, Wu CT, Young DL (2014) A novel upwind-based local radial basis function differential quadrature method for convection-dominated flows. Comput Fluids 89:157–166

    MathSciNet  MATH  Google Scholar 

  • Chen CS, Fan CM, Wen PH (2011) The method of approximate particular solutions for solving elliptic problems with variable coefficients. Int J Comput Methods 8(3):545–559

    MathSciNet  MATH  Google Scholar 

  • Chen W, Fu Z, Chen C (2014) Recent advances in radial basis function collocation methods. Springer Briefs in Applied Sciences and Technology, Berlin

    MATH  Google Scholar 

  • Cheng AHD, Golberg MA, Kansa EJ, Zammito G (2003) Exponential convergence and \(h-c\) multiquadric collocation method for partial differential equations. Numer Methods Part Differ Equ 19(5):571–594

    MathSciNet  MATH  Google Scholar 

  • Dag I, Irk D, Saka B (2005) A numerical solution of the Burgers’ equation using cubic B-splines. Appl Math Comput 163:199–211

    MathSciNet  MATH  Google Scholar 

  • Dag I, Saka B, Boz A (2005) B-spline Galerkin methods for numerical solutions of the Burgers’ equation. Appl Math Comput 166:506–522

    MathSciNet  MATH  Google Scholar 

  • Dormand JR, Prince PJ (1980) A family of embedded Runge–Kutta formulae. J Comput Appl Math 6:19–26

    MathSciNet  MATH  Google Scholar 

  • Fasshauer GE (2007) Meshfree approximation methods with MATLAB, River Edge, NJ. World Scientific, Hackensack

    Google Scholar 

  • Fasshauer GE, McCourt M (2015) Kernel-based approximation methods using MATLAB. World Scientific, Hackensack

    MATH  Google Scholar 

  • Fornberg B, Piret C (2008) On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere. J Comput Phys 227:2758–2780

    MathSciNet  MATH  Google Scholar 

  • Ganaie I, Kukreja V (2014) Numerical solution of Burgers’ equation by cubic Hermite collocation method. Appl Math Comput 237:571–581

    MathSciNet  MATH  Google Scholar 

  • Golbabai A, Mohebianfar E, Rabiei H (2015) On the new variable shape parameter strategies for radial basis functions. Comput Appl Math 34:691–704

    MathSciNet  MATH  Google Scholar 

  • Gottlieb S, Ketcheson D, Shu C-W (2011) Strong stability preserving Runge–Kutta and multistep time discretizations. World Scientific, Hackensack

    MATH  Google Scholar 

  • Guo Y, Shi Y-F, Li Y-M (2016) A fifth-order finite volume weighted compact scheme for solving one-dimensional Burgers’ equation. Appl Math Comput 281:172–185

    MathSciNet  MATH  Google Scholar 

  • Haq S, Ul-Islam S, Uddin M (2009) A mesh-free method for the numerical solution of the KdV–Burgers equation. Appl Math Model 33:3442–3449

    MathSciNet  MATH  Google Scholar 

  • Haq S, Hussain A, Uddin M (2011) On the numerical solution of nonlinear Burgers-type equations using meshless method of lines. Appl Math Comput 218:6280–6290

    MathSciNet  MATH  Google Scholar 

  • Hassanien I, Salama A, Hosham H (2005) Fourth-order finite difference method for solving Burgers’ equation. Appl Math Comput 170:781–800

    MathSciNet  MATH  Google Scholar 

  • Hon YC, Mao XZ (1998) An efficient numerical scheme for Burgers’ equation. Appl Math Comput 95:37–50

    MathSciNet  MATH  Google Scholar 

  • Hosseini B, Hashemi R (2011) Solution of Burgers’ equation using a local-RBF meshless method. Int J Comput Methods Eng Sci Mech 12(1):44–58

    MathSciNet  MATH  Google Scholar 

  • Hussain M, Haq S (2019) A computational study of solitary waves solution of Kawahara-type equations by meshless spectral interpolation method. Int J Mod Phys C 30(12):1950102

    MathSciNet  Google Scholar 

  • Hussain M, Haq S (2020) Numerical simulation of solitary waves of Rosenau–KdV equation by Crank–Nicolson meshless spectral interpolation method. Eur Phys J Plus 35:98. https://doi.org/10.1140/epjp/s13360-020-00156-7

    Article  Google Scholar 

  • Hussain M, Haq S (2020) A hybrid RBFs collocation technique to numerically solve fractional advection–diffusion models. Numer Methods Part Differ Equ 36(6):1254–1279

    Google Scholar 

  • Hussain M, Haq S, Ghafoor A, Ali I (2020) Numerical solutions of time-fractional coupled viscous Burgers’ equations using meshfree spectral method. Comput Appl Math 39:6

    MathSciNet  MATH  Google Scholar 

  • Jiwari R (2012) A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation. Comput Phys Commun 183(11):2413–2423

    MathSciNet  MATH  Google Scholar 

  • Jiwari R (2015) A hybrid numerical scheme for the numerical solution of the Burgers’ equation. Comput Phys Commun 188:59–67

    MathSciNet  MATH  Google Scholar 

  • Jiwari R, Mittal R, Sharma KK (2013) A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers’ equation. Appl Math Comput 219(12):6680–6691

    MathSciNet  MATH  Google Scholar 

  • Jiwari R, Kumar S, Mittal RC (2019) Meshfree algorithms based on radial basis functions for numerical simulation and to capture shocks behavior of Burgers’ type problems. Eng Comput 36(4):1142–1168

    Google Scholar 

  • Jiwari R, Kumar S, Mittal RC, Awrejcewicz J (2020) A meshfree approach for analysis and computational modeling of non-linear Schrödinger equation. Comput Appl Math 39:95

    MATH  Google Scholar 

  • Kansa EJ (1990) Multiquadrics—a scattered data approximation scheme with application to computation fluid dynamics, II. Solutions to hyperbolic, parabolic, and elliptic partial differential equations. Comput Math Appl 19:149–161

    MathSciNet  Google Scholar 

  • Kansa EJ, Holoborodko P (2017) On the ill-conditioned nature of \(C^{\infty }\) RBF strong collocation. Eng Anal Bound Elem 78:26–30

    MathSciNet  MATH  Google Scholar 

  • Khater AH, Temsah RS, Hassan MM (2008) A Chebyshev spectral collocation method for solving Burgers’-type equations. J Comput Appl Math 222:333–350

    MathSciNet  MATH  Google Scholar 

  • Korkmaz A, Dag I (2011) Polynomial based differential quadrature method for numerical solution of nonlinear Burgers’ equation. J Frankl Inst 348(10):2863–2875

    MathSciNet  MATH  Google Scholar 

  • Larsson E, Fornberg B (2003) A numerical study of some radial basis function based solution methods for elliptic PDEs. Comput Math Appl 46:891–902

    MathSciNet  MATH  Google Scholar 

  • Larsson E, Sundin U (2020) An investigation of global radial basis function collocation methods applied to Helmholtz problems. arXiv:2001.11090v1

  • LeVeque RJ (2007) Finite difference methods for ordinary and partial differential equations. SIAM Publications, Philadelphia

    MATH  Google Scholar 

  • Li J, Cheng AHD, Chen CS (2003) A comparison of efficiency and error convergence of multiquadric collocation method and finite element method. Eng Anal Bound Elem 27:251–257

    MATH  Google Scholar 

  • Liao W (2008) An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation. Appl Math Comput 206(2):755–764

    MathSciNet  MATH  Google Scholar 

  • Liu GR, Gu YT (2005) An introduction to meshfree methods and their programming. Springer, Dordrecht

    Google Scholar 

  • Martinez FMB (2008) Meshless methods for elliptic and free boundary problems. PhD Thesis, Universidad Carlos III De Madrid

  • Mishra PK, Nath SK, Kosec G, Sen MK (2017) An improved radial basis-pseudospectral scheme with hybrid Gaussian-cubic kernels. Eng Anal Bound Elem 80:162–171

    MathSciNet  MATH  Google Scholar 

  • Mishra PK, Nath SK, Sen MK, Fasshauer GE (2018) Hybrid Gaussian-cubic radial basis functions for scattered data interpolation. Comput Geosci 22(5):1203–1218

    MathSciNet  MATH  Google Scholar 

  • Mishra PK, Fasshauer GE, Sen MK, Ling L (2019) A stabilized radial basis-finite difference (RBF-FD) method with hybrid kernels. Comput Math Appl 77(9):2354–2368. https://doi.org/10.1016/j.camwa.2018.12.027

    Article  MathSciNet  MATH  Google Scholar 

  • Mittal R, Jain R (2012) Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method. Appl Math Comput 218:7839–7855

    MathSciNet  MATH  Google Scholar 

  • Mittal R, Jiwari R (2012) A differential quadrature method for numerical solutions of Burgers’-type equations. Int J Numer Methods Heat Fluid Flow 22(7):880–895

    MathSciNet  MATH  Google Scholar 

  • Mittal RC, Rohila R (2017) A study of one dimensional nonlinear diffusion equations by Bernstein polynomial based differential quadrature method. J Math Chem 55(2):673–695

    MathSciNet  MATH  Google Scholar 

  • Mittal R, Jiwari R, Sharma KK (2012) A numerical scheme based on differential quadrature method to solve time dependent Burgers’ equation. Eng Comput 30(1):117–131

    Google Scholar 

  • Mukundan V, Awasthi A (2015) Efficient numerical techniques for Burgers’ equation. Appl Math Comput 262:282–297

    MathSciNet  MATH  Google Scholar 

  • Platte R, Driscoll T (2006) Eigenvalue stability of radial basis functions discretizations for time-dependent problems. Comput Math Appl 51:1251–1268

    MathSciNet  MATH  Google Scholar 

  • Ramadam MA, Danaf TE, Alaal FA (2005) A numerical solution of the Burgers’ equation using septic B-splines. Chaos Soliton Fract 26:795–804

    MathSciNet  MATH  Google Scholar 

  • Rashidinia J, Rasoulizadeh MN (2019) Numerical methods based on radial basis function-generated finite difference (RBF-FD) for solution of GKdVB equation. Wave Motion 90:152–167

    MathSciNet  MATH  Google Scholar 

  • Raslan KR (2003) A collocation solution for Burgers’ equation using quadratic B-spline finite elements. Int J Comput Math 80:931–938

    MathSciNet  MATH  Google Scholar 

  • Saka B, Dag I (2007) Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation. Chaos Soliton Fract 32:1125–1137

    MathSciNet  MATH  Google Scholar 

  • Sari M, Gurarslan G (2009) A sixth-order compact finite difference scheme to the numerical solutions of Burgers’ equation. Appl Math Comput 208:475–483

    MathSciNet  MATH  Google Scholar 

  • Sarra SA, Kansa EJ (2009) Multiquadric radial basis function approximation methods for the numerical solution of partial differential equations. Adv Comput Mech

  • Sarra SA (2012) A local radial basis function method for advection–diffusion–reaction equations on complexly shaped domains. Appl Math Comput 218:9853–9865

    MathSciNet  MATH  Google Scholar 

  • Schaback R (1995) Error estimates and condition numbers for radial basis function interpolation. Adv Comput Math 3:389–396

    MathSciNet  MATH  Google Scholar 

  • Schiesser WE (1991) The numerical method of lines: integration of partial differential equations. Academic Press, San Diego

    MATH  Google Scholar 

  • Schiesser WE, Griffiths GW (2009) A compendium of partial differential equation models: method of lines analysis with Matlab. Cambridge University Press, New York

    MATH  Google Scholar 

  • Shao L, Feng X, He Y (2011) The local discontinuous Galerkin finite element method for Burgers’ equation. Math Comput Model 54(11):2943–2954

    MathSciNet  MATH  Google Scholar 

  • Shivanian E, Abbasbandy S (2020) Pseudospectral meshless radial point interpolation for generalized biharmonic equation in the presence of Cahn–Hilliard conditions. Comput Appl Math 39:148

    MathSciNet  MATH  Google Scholar 

  • Shu C, Ding H, Yeo S (2003) Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 192:941–954

    MATH  Google Scholar 

  • Shu C, Ding H, Yeo S (2005) Computation of incompressible Navier–Stokes equations by local RBF based differential quadrature method. Tech Sci Press CMES 7(2):195–205

    MathSciNet  MATH  Google Scholar 

  • Trefethen LN (2000) Spectral methods in MATLAB. SIAM Publications, Philadelphia

    MATH  Google Scholar 

  • Xu M, Wang R-H, Zhang J-H, Fang Q (2011) A novel numerical scheme for solving Burgers’ equation. Appl Math Comput 217:4473–4482

    MathSciNet  MATH  Google Scholar 

  • Zhang P-G, Wang J-P (2012) A predictor–corrector compact finite difference scheme for Burgers’ equation. Appl Math Comput 219(3):892–898

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank the editor and anonymous reviewers of this manuscript whose suggestions greatly improved the presentation of this manuscript.

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  1. Department of Mathematics, Faculty of Science and Technology, Women University of Azad Jammu and Kashmir, Bagh, Azad Kashmir, Pakistan

    Manzoor Hussain

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  1. Manzoor Hussain
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Hussain, M. Hybrid radial basis function methods of lines for the numerical solution of viscous Burgers’ equation. Comp. Appl. Math. 40, 107 (2021). https://doi.org/10.1007/s40314-021-01505-7

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