Fourth-order compact schemes for the numerical simulation of coupled Burgers' equation

Abstract This paper introduces two new modified fourth-order exponential time differencing Runge–Kutta (ETDRK) schemes in combination with a global fourth-order compact finite difference scheme (in space) for direct integration of nonlinear coupled viscous Burgers’ equations in their original form without using any transformations or linearization techniques. One scheme is a modification of the Cox and Matthews ETDRK4 scheme based on ( 1 , 3 ) -Pade approximation and other is a modification of Krogstad’s ETDRK4-B scheme based on ( 2 , 2 ) -Pade approximation. Efficient versions of the proposed schemes are obtained by using a partial fraction splitting technique of rational functions. The stability properties of the proposed schemes are studied by plotting the stability regions, which provide an explanation of their behavior for dispersive and dissipative problems. The order of convergence of the schemes is examined empirically and found that the modification of ETDRK4 converges with the expected rate even if the initial data are nonsmooth. On the other hand, modification of ETDRK4-B suffers with order reduction if the initial data are nonsmooth. Several numerical experiments are carried out in order to demonstrate the performance and adaptability of the proposed schemes. The numerical results indicate that the proposed schemes provide better accuracy than other schemes available in the literature. Moreover, the results show that the modification of ETDRK4 is reliable and yields more accurate results than modification of ETDRK4-B, while solving problems with nonsmooth data or with high Reynolds number.

[1]  Bengt Fornberg,et al.  Classroom Note: Calculation of Weights in Finite Difference Formulas , 1998, SIAM Rev..

[2]  Lloyd N. Trefethen,et al.  Fourth-Order Time-Stepping for Stiff PDEs , 2005, SIAM J. Sci. Comput..

[3]  Hillel Tal-Ezer,et al.  On Restart and Error Estimation for Krylov Approximation of w=f(A)v , 2007, SIAM J. Sci. Comput..

[4]  A. Y. Suhov.  A Spectral Method for the Time Evolution in Parabolic Problems , 2006, J. Sci. Comput..

[5]  Weizhong Dai,et al.  Fourth‐order compact schemes of a heat conduction problem with Neumann boundary conditions , 2007 .

[6]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[7]  Paul Arminjon,et al.  Numerical solution of burgers' equations in two space dimensions☆ , 1979 .

[8]  Wenyuan Liao,et al.  An implicit fourth-order compact finite difference scheme for one-dimensional Burgers' equation , 2008, Appl. Math. Comput..

[9]  C. Lubich,et al.  On Krylov Subspace Approximations to the Matrix Exponential Operator , 1997 .

[10]  Ram Jiwari,et al.  A differential quadrature method for numerical solutions of Burgers'‐type equations , 2012 .

[11]  Weizhong Dai,et al.  Compact operator method of accuracy two in time and four in space for the numerical solution of coupled viscous Burgers' equations , 2015, Appl. Math. Comput..

[12]  Abdul-Qayyum M. Khaliq,et al.  The locally extrapolated exponential time differencing LOD scheme for multidimensional reaction-diffusion systems , 2015, J. Comput. Appl. Math..

[13]  E. Hopf.  The partial differential equation ut + uux = μxx , 1950 .

[14]  Jianping Zhu,et al.  Efficient and accurate finite difference schemes for solving one-dimensional Burgers’ equation , 2011, Int. J. Comput. Math..

[15]  Esipov.  Coupled Burgers equations: A model of polydispersive sedimentation. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  Ram Jiwari,et al.  A hybrid numerical scheme for the numerical solution of the Burgers' equation , 2015, Comput. Phys. Commun..

[17]  M. Yousuf,et al.  Smoothing schemes for reaction-diffusion systems with nonsmooth data , 2009 .

[18]  S. Cox,et al.  Exponential Time Differencing for Stiff Systems , 2002 .

[19]  Donald J. Kouri,et al.  Distributed approximating functional approach to Burgers' equation in one and two space dimensions , 1998 .

[20]  J. M. Keiser,et al.  A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs , 1998 .

[21]  Do ˘ Kaya,et al.  An explicit solution of coupled viscous Burgers' equation by the decomposition method , 2001 .

[22]  Harpreet Kaur,et al.  Haar wavelet-based numerical investigation of coupled viscous Burgers' equation , 2015, Int. J. Comput. Math..

[23]  Geeta Arora,et al.  Numerical solution of the coupled viscous Burgers’ equation , 2011 .

[24]  A. Ostermann,et al.  Implementation of exponential Rosenbrock-type integrators , 2009 .

[25]  S. Krogstad.  Generalized integrating factor methods for stiff PDEs , 2005 .

[26]  Jichao Zhao,et al.  Highly accurate compact mixed methods for two point boundary value problems , 2007, Appl. Math. Comput..

[27]  T. Driscoll,et al.  Regular Article: A Fast Spectral Algorithm for Nonlinear Wave Equations with Linear Dispersion , 1999 .

[28]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[29]  Ahmad Izani Bin Md. Ismail,et al.  A Fourier Pseudospectral Method for Solving Coupled Viscous Burgers Equations , 2009, Comput. Methods Appl. Math..

[30]  Siraj-ul-Islam,et al.  A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations , 2008 .

[31]  Manoj Kumar,et al.  A composite numerical scheme for the numerical simulation of coupled Burgers' equation , 2014, Comput. Phys. Commun..

[32]  E. H. Twizell,et al.  On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal Padé approximations , 1993 .

[33]  Raimund Bürger,et al.  Linearly Implicit IMEX Runge-Kutta Methods for a Class of Degenerate Convection-Diffusion Problems , 2015, SIAM J. Sci. Comput..

[34]  A. A. Soliman,et al.  Variational iteration method for solving Burger's and coupled Burger's equations , 2005 .

[35]  S. Kutluay,et al.  Numerical solution of one-dimesional Burgers equation: explicit and exact-explicit finite difference methods , 1999 .

[36]  A. H. Khater,et al.  A Chebyshev spectral collocation method for solving Burgers'-type equations , 2008 .

[37]  Abdou,et al.  VARIATIONAL ITERATION METHOD FOR SOLVING BURGERS AND COUPLED BURGERS EQUATIONS , 2005 .

[38]  J. Cole.  On a quasi-linear parabolic equation occurring in aerodynamics , 1951 .

[39]  Abdul-Qayyum M. Khaliq,et al.  Higher order exponential time differencing scheme for system of coupled nonlinear Schrödinger equations , 2014, Appl. Math. Comput..

[40]  Qiang Du,et al.  Analysis and Applications of the Exponential Time Differencing Schemes and Their Contour Integration Modifications , 2005 .

[41]  Muhammad Abbas,et al.  Numerical solution of the coupled viscous Burgers equations by Chebyshev-Legendre Pseudo-Spectral method , 2014, Appl. Math. Comput..

[42]  A. A. Soliman,et al.  The modified extended tanh-function method for solving Burgers-type equations , 2006 .

[43]  S. P. Nørsett,et al.  Attainable order of rational approximations to the exponential function with only real poles , 1977 .