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A spatial sixth-order CCD-TVD method for solving multidimensional coupled Burgers’ equation

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Abstract

In this paper, a high-order compact difference scheme is proposed for solving multidimensional nonlinear Burgers’ equation. The three-stage third-order total variation diminishing (TVD) Runge–Kutta scheme is employed in time, and the three-point combined compact difference (CCD) scheme is used for spatial discretization. The proposed TVD-CCD method is free of using Hopf–Cole transformation, and treats the nonlinear term explicitly. Thus it is very efficient and easy to implement. Our method is effective to capture shock wave, third-order accurate in time, and sixth-order accurate in space. In addition, we show the unique solvability of the CCD system under non-periodic boundary conditions. Numerical experiments are given to demonstrate the high efficiency and accuracy of the proposed method.

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References

  • Bahadir AR (2003) A fully implicit finite-difference scheme for two-dimensional Burgers’ equations. Appl Math Comput 137:131–137

    MathSciNet  MATH  Google Scholar 

  • Bahadir AR, Saǧlam M (2005) A mixed finite difference and boundary element approach to one-dimensional Burgers’ equation. Appl Math Comput 160:663–673

    MathSciNet  MATH  Google Scholar 

  • Bateman H (1914) Some recent researches on the motion of fluids. Mon Weather Rev 43:163–170

    Google Scholar 

  • Caldwell J, Wanless P, Cook AE (1981) A finite element approach to Burgers’ equation. Appl Math Model 5:189–193

    MathSciNet  MATH  Google Scholar 

  • Campos MD, Romão EC (2014) A high-order finite-difference scheme with a linearization technique for solving of three-dimensional Burgers’ equation. CMES-Comp Model Eng 103:139–154

    MathSciNet  MATH  Google Scholar 

  • Chen B, He D, Pan K (2018) A linearized high-order combined compact difference scheme for multi-dimensional coupled Burgers’ equations. Numer Math Theor Methods Appl 11:299–320

    MathSciNet  MATH  Google Scholar 

  • Chen B, He D, Pan K (2019) A CCD-ADI method for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients. Int J Comput Math 96:992–1004

    MathSciNet  Google Scholar 

  • Chu P, Fan C (1998) A three-point combined compact difference scheme. J Comput Phys 140:370–399

    MathSciNet  MATH  Google Scholar 

  • Cole JD (1951) On a quasilinear parabolic equation occurring in aerodynamics. Quart Appl Math 9:225–236

    MathSciNet  MATH  Google Scholar 

  • Davidson GA (1975) A Burgers’ equation approach to finite amplitude acoustics in aerosol media. J Sound Vib 38:475–495

    MATH  Google Scholar 

  • Dehghan M, Saray BN, Lakestani M (2014) Mixed finite difference and Galerkin methods for solving Burgers equations using interpolating scaling functions. Math Methods Appl Sci 37:894–912

    MathSciNet  MATH  Google Scholar 

  • Esipov SE (1995) Coupled Burgers’ equations: a model of poly-dispersive sedimentation. Phys Rev E 52:3711–3718

    Google Scholar 

  • Fletcher CA (1983) Generating exact solutions of the two-dimensional Burgers’ equations. Int J Numer Methods Fluids 3:213–216

    MATH  Google Scholar 

  • Gao G, Sun H (2015) Three-point combined compact difference schemes for time-fractional advection-diffusion equations with smooth solutions. J Comput Phys 298:520–538

    MathSciNet  MATH  Google Scholar 

  • Gao Q, Zou M (2017) An analytical solution for two and three dimensional nonlinear Burgers’ equation. Appl Mathods Model 45:255–270

    MathSciNet  MATH  Google Scholar 

  • Gottlieb S, Shu C (1998) Total variation diminishing Runger–Kutta schemes. Math Comput 221:73–85

    MATH  Google Scholar 

  • Gülsu M (2006) A finite difference approach for solution of Burgers’ equation. Appl Math Comput 175:1245–1255

    MathSciNet  MATH  Google Scholar 

  • Gülsu M, Özis T (2005) Numerical solution of Burgers’ equation with restrictive Taylor approximations. Appl Math Comput 171:1192–1200

    MathSciNet  MATH  Google Scholar 

  • He D (2016) An unconditionally stable spatial sixth-order CCD-ADI method for the two-dimensional linear hyperbolic equation. Numer Algorithms 72:1103–1117

    MathSciNet  MATH  Google Scholar 

  • He D, Pan K (2017) An unconditionally stable linearized CCD-ADI method for generalized nonlinear Schrödinger equations with variable coefficients in two and three dimensions. Comput Math Appl 73:2360–2374

    MathSciNet  MATH  Google Scholar 

  • He D, Pan K (2018) A fifth-order combined compact difference scheme for the Stokes flow on polar geometries. E Asian J Appl Math 7:714–727

    MathSciNet  MATH  Google Scholar 

  • He D, Pan K (2018) An unconditionally stable linearized difference scheme for the fractional Ginzburg–Landau equation. Numer Algorithms 79:899–925

    MathSciNet  MATH  Google Scholar 

  • Hopf E (1950) The partial differential equation \(u_t+uu_x=\mu u_{xx}\). Commun Pure Appl Math 3:201–230

    MATH  Google Scholar 

  • Huang P, Abduwali A (2010) The Modified Local Crank–Nicolson method for one- and two-dimensional Burgers’ equations. Comput Math Appl 59:2452–2463

    MathSciNet  MATH  Google Scholar 

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

    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 

  • Khesin B, Misiolek G (2007) Shock waves for the Burgers equation and curvatures of diffeomorphism groups. Proc Steklov I Math 259:73–81

    MathSciNet  MATH  Google Scholar 

  • Kraenkel RA, Pereira JG, Manna MA (1992) Nonlinear surface-wave excitations in the Benard–Marangoni system. Phys Rev A 46:4786–4790

    Google Scholar 

  • Kumar M, Pandit S (2014) A composite scheme for the numerical simulation of coupled Burgers’ equation. Comput Phys Commun 185:809–817

    MathSciNet  MATH  Google Scholar 

  • Kutluay S, Bahadir AR (1999) Numerical solution of one-dimensional Burgers’ equation: explicit and exact-explicit finite difference methods. J Comput Appl Math 103:251–261

    MathSciNet  MATH  Google Scholar 

  • Kutluay S, Ucar Y (2013) Numerical solutions of the coupled Burgers’ equation by the Galerkin quadratic B-spline finite element method. Math Methods Appl Sci 36:2403–2415

    MathSciNet  MATH  Google Scholar 

  • Lee S, Liu J, Sun H (2014) Combined compact difference scheme for linear second-order partial differential equations with mixed derivative. J Comput Appl Math 264:23–37

    MathSciNet  MATH  Google Scholar 

  • Lewis R, Nithiarasu P, Seetharamu K (2004) Fundamentals of the finite element method for heat and fluid flow. John Wiley & Sons

  • Li L, Sun H, Tam S (2015) A spatial sixth-order alternating direction implicit method for two-dimensional cubic nonlinear Schrödinger equations. Comput Phys Commun 187:38–48

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Liao W (2010) A fourth-order finite-difference method for solving the system of two-dimensional Burgers’ equations. Int J Numer Methods Fluid 64:565–590

    MathSciNet  MATH  Google Scholar 

  • Liao W, Zhu J (2011) Efficient and accurate finite difference schemes for solving one-dimensional Burgers’ equation. Int J Comput Math 88:2575–2590

    MathSciNet  MATH  Google Scholar 

  • Mahesh K (1998) A family of high order finite difference schemes with good spectral resolution. J Comput Phys 145:332–358

    MathSciNet  MATH  Google Scholar 

  • Mittal RC, Jiwari R (2012) A differential quadrature method for solving Burgers’-type equation. Int J Numer Methods Heat Fluids Flow 22:880–895

    MATH  Google Scholar 

  • Pan K, Jin X, He D (2020) Pointwise error estimates of a linearized difference scheme for strongly coupled fractional Ginzburg-Landau equations. Math Methods Appl Sci 43:512–535

    Google Scholar 

  • Sari M, Gürarslan 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 

  • 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 

  • Shandarin SF (1997) Three dimensional Burgers’ equation as a model for the Large-scale structure Formation in the Universe. IMA 85:401–413

    MathSciNet  MATH  Google Scholar 

  • Shukl HS, Tamsir M et al (2016) Modified cubic B-spline differential quadrature method for numerical solution of three-dimensional coupled viscous Burgers’ equation. Mod Phys Lett B 30:1650110

    MathSciNet  Google Scholar 

  • Su NH, Watt PC et al (2004) Analysis of turbulent flow patterns of soil water under filed conditions using Burgers’ equation and porous suction-cup samplers. Aust J Soil Res 42:9–16

    Google Scholar 

  • Sun H, Li L (2014) A CCD-ADI method for unsteady convection-diffusion equations. Comput Phys Commun 185:790–797

    MathSciNet  MATH  Google Scholar 

  • Tamsir M, Srivastava VK, Jiwari R (2016) An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation. Appl Math Comput 290:111–124

    MathSciNet  MATH  Google Scholar 

  • Varöglu E, Finn WDL (1980) Space-time finite elements incorporating characteristics for the Burgers’ equation. Int J Numer Methods Eng 16:171–184

    MathSciNet  MATH  Google Scholar 

  • Wang QH, Pan KJ, Hu HL (2018) Unique solvability of the CCD scheme for convectionCdiffusion equations with variable convection coefficients. Adv Differ Equ 2018:163

    MATH  Google Scholar 

  • Xie S, Li G et al (2010) A compact finite difference method for solving Burgers’ equation. Int J Numer Methods Fluids 62:747–764

    MathSciNet  MATH  Google Scholar 

  • Yadav OP, Jiwari R (2017) Finite element analysis and approximation of Burgers–Fisher equation. Numer Methods Part Differ Equ 33:1652–1677

    MathSciNet  MATH  Google Scholar 

  • Yang L, Pu X (2016) Derivation of the Burgers’ equation from the gas dynamics. Commun Math Sci 14:671–682

    MathSciNet  MATH  Google Scholar 

  • Yue X, Bu W et al (2018) Fully finite element adaptive AMG method for time-space Caputo–Riesz fractional diffusion equations. Adv Appl Math Mech 10:1103–1125

    MathSciNet  Google Scholar 

  • Yue X, Shu S et al (2019) Parallel-in-time multigrid for space-time finite element approximations of two-dimensional space-fractional diffusion equations. Comput Math Appl 78:3471–3484

    MathSciNet  Google Scholar 

  • Yue X, Liu M et al (2019) Space-time finite element adaptive AMG for multi-term time fractional advection diffusion equations. Math Methods Appl Sci. https://doi.org/10.1002/mma.5876

  • Zabusky NJ, Kruskal MD (1965) Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243

    MATH  Google Scholar 

  • Zhu H, Shu H, Ding M (2010) Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method. Comput Math Appl 60:840–848

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank the anonymous editor and reviewers for their constructive comments that improved the paper substantially. Kejia Pan was supported by Science Challenge Project (TZ2016002), the Natural Science Foundation of China (41874086), the Excellent Youth Foundation of Hunan Province of China (2018JJ1042) and the Innovation-Driven Project of Central South University (2018CX042). Xiaoqiang Yue was supported by the Natural Science Foundation of China (11601462, 11901189), Project of Scientific Research Fund of Hunan Provincial Science and Technology Department (2018WK4006) and Hunan Provincial Natural Science Foundation of China (2018JJ3494), Xiaoxin Wu was supported by Tian’an Cup College Students’ innovation and entrepreneurship training Project (TAB2019-01).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, China

    Kejia Pan, Xiaoxin Wu & Runxin Ni

  2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, China

    Xiaoqiang Yue

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  1. Kejia Pan
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Correspondence to Xiaoqiang Yue.

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Appendices

Appendix A.

$$\begin{aligned} T_1&= -\frac{136835}{8209824}+\frac{421733\sqrt{7}}{36944208} ,\\ T_2&= -\frac{416144963942525\sqrt{7}}{864691128455135232} ,\\ T_3&= -\frac{46840306656146665409\sqrt{7}}{41595480345574524321792}+\frac{6115}{2052456} ,\\ T_4&= -\frac{46840306656146665409\sqrt{7}}{41595480345574524321792}+\frac{6115}{2052456} ,\\ T_5&= \frac{21881309630676858473952943462656048141172736\sqrt{7}}{4667640113605791995493311956355581953955543731} \\&\quad -\frac{2893014312251833953326629616913521171234816}{518626679289532443943701328483953550439504859} , \\ T_6&= -\frac{267591658885243284604171740430098010027602763}{33192107474530076412396885022973027228128310976}\\&\quad +\frac{587650275369111747907115169185577623944244693\sqrt{7}}{37341120908846335963946495650844655631644349848} ,\\ T_7&= -\frac{1799614987336256538927773374693436599301849447\sqrt{7}}{298728967270770687711571965206757245053154798784} \\ {}&\quad -\frac{442381615984325718712039486584685244485625}{691502239052709925258268437978604733919339812} ,\\ T_8&= \frac{60812707732120381749986704242152551792357469\sqrt{7}}{18670560454423167981973247825422327815822174924}\\&\quad +\frac{900723013413257827633193252372996530470617}{922002985403613233677691250638139645225786416} ,\\ T_9&= \frac{5235921989442888980950159\sqrt{7}}{183849407004430256490676224}+\frac{41853249163690425155625}{40855423778762279220150272} ,\\ T_{10}&= \frac{640001231916311360619949\sqrt{7}}{551548221013290769472028672}+\frac{33366161622715196997673}{40855423778762279220150272} ,\\ T_{11}&= \frac{69251\sqrt{7}}{2574720}+\frac{133}{85824} ,\\ T_{13}&= -\frac{\sqrt{7}}{17280} ,\\ T_{14}&= \frac{34711\sqrt{7}}{8582400}+\frac{7073}{1430400} ,\\ T_{15}&= -\frac{3197\sqrt{7}}{1029888}+\frac{2315}{85824} ,\\ T_{16}&= -\frac{4733\sqrt{7}}{2574720}+\frac{479}{107280} ,\\ T_{17}&= -\frac{547\sqrt{7}}{80460}-\frac{29}{26820} ,\\ T_{18}&= \frac{2711\sqrt{7}}{16092}-\frac{1495}{5364} ,\\ T_{19}&= \frac{547\sqrt{7}}{80460}+\frac{29}{26820} . \end{aligned}$$

Appendix B. Matlab code for deriving (28)

figure b

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Pan, K., Wu, X., Yue, X. et al. A spatial sixth-order CCD-TVD method for solving multidimensional coupled Burgers’ equation. Comp. Appl. Math. 39, 76 (2020). https://doi.org/10.1007/s40314-020-1063-6

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