The Laplace transform method for Burgers' equation

• Published in: International journal for numerical methods in fluids Vol. 63; no. 9; pp. 1060 - 1076
• Main Authors: Chen, Suqin, Wu, Xionghua, Wang, Yingwei, Kong, Weibin
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 30-07-2010; Wiley
• Subjects: Burgers' equation; Chebyshev approximation; Computational methods in fluid dynamics; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); increment linearization technique; Inversions; Laplace transforms; Linearization; Mathematical analysis; Mathematical models; Numerical analysis; Partial differential equations; Physics; rational collocation method; Talbot's method and numerical inversion; the Laplace transform; Computational fluid dynamics; increment linearization technique; Digital simulation; Boundary conditions; the Laplace transform; Burgers equation; Shock waves; Talbot's method and numerical inversion; Modelling; Laplace transformation; Burgers' equation; rational collocation method; Collocation method; Linearization
• ISSN: 0271-2091; 1097-0363; 1097-0363
• DOI: 10.1002/fld.2116

The Laplace transform method (LTM) is introduced to solve Burgers' equation. Because of the nonlinear term in Burgers' equation, one cannot directly apply the LTM. Increment linearization technique is introduced to deal with the situation. This is a key idea in this paper. The increment linearization technique is the following: In time level t, we divide the solution u(x, t) into two parts: u(x, tk) and w(x, t), tk⩽t⩽tk+1, and obtain a time‐dependent linear partial differential equation (PDE) for w(x, t). For this PDE, the LTM is applied to eliminate time dependency. The subsequent boundary value problem is solved by rational collocation method on transformed Chebyshev points. To face the well‐known computational challenge represented by the numerical inversion of the Laplace transform, Talbot's method is applied, consisting of numerically integrating the Bromwich integral on a special contour by means of trapezoidal or midpoint rules. Numerical experiments illustrate that the present method is effective and competitive. Copyright © 2009 John Wiley & Sons, Ltd.