Numerical solution of the Burgers' equation by automatic differentiation

Abstract We compute the solution of the one-dimensional Burgers’ equation by marching the solution in time using a Taylor series expansion. Our approach does not require symbolic manipulation and does not involve the solution of a system of linear or non-linear algebraic equations. Instead, we use recursive formulas obtained from the differential equation to calculate exact values of the derivatives needed in the Taylor series. We illustrate the effectiveness of our method by solving four test problems with known exact solutions. The numerical solutions we obtain are in excellent agreement with the exact solutions, while being superior to other previously reported numerical solutions.

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

[2]  George W. Platzman,et al.  A table of solutions of the one-dimensional Burgers equation , 1972 .

[3]  Turgut Öziş,et al.  A direct variational methods applied to Burgers' equation , 1996 .

[4]  M. Abd-el-Malek,et al.  Group theoretic methods applied to Burgers' equation , 2000 .

[5]  David J. Evans,et al.  The Group Explicit method for the solution of Burger's equation , 1984, Computing.

[6]  Poonam Singhal,et al.  Numerical solution of Burger's equation , 1993 .

[7]  J. Burgers.  A mathematical model illustrating the theory of turbulence , 1948 .

[8]  H. Bateman,et al.  SOME RECENT RESEARCHES ON THE MOTION OF FLUIDS , 1915 .

[9]  D. J. Evans,et al.  The numerical solution of burgers' equation by the alternating group explicit (age) method , 1989 .

[10]  Cedric Taylor,et al.  Numerical Methods for Non-linear Problems , 1986 .

[11]  Andreas Griewank,et al.  Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition , 2000, Frontiers in applied mathematics.

[12]  Peter Smith,et al.  Solution of Burgers' equation with a large Reynolds number , 1982 .

[13]  Abdulkadir Dogan,et al.  A least‐squares finite element scheme for the RLW equation , 1996 .

[14]  Louis B. Rall,et al.  Automatic Differentiation: Techniques and Applications , 1981, Lecture Notes in Computer Science.

[15]  W. L. Wood.  An exact solution for Burger's equation , 2006 .

[16]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.

[17]  L. Gardner,et al.  A collocation solution for Burgers' equation using cubic B-spline finite elements , 1992 .

[18]  E. N. Aksan,et al.  A numerical solution of Burgers' equation , 2004, Appl. Math. Comput..

[19]  Mohamed A. Ramadan,et al.  Application of the Non-Polynomial Spline Approach to the Solution of the Burgers' Equation , 2007 .

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

[21]  Kazuhiko Kakuda,et al.  The generalized boundary element approach to Burgers' equation , 1990 .

[22]  J. Reynen,et al.  A space-time least-square finite element scheme for advection-diffusion equations , 1984 .

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

[24]  W. D. Liam Finn,et al.  Space‐time finite elements incorporating characteristics for the burgers' equation , 1980 .

[25]  Alaattin Esen,et al.  Numerical solutions of the Burgers' equation by the least-squares quadratic B-spline finite element method , 2004 .