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A multi-domain Galerkin method is presented for the Burgers equation, whose solutions behave differently at positive infinity and negative infinity. The original problem is transformed into a problem with general variable coefficients and homogeneous boundary conditions at infinities by using the auxiliary function, which is capable of being reformulated as a suitable variational formulation that is the most convenient for analysing numerical error. As an application, the composite spectral scheme with numerical integration is provided for the underlying problem on the whole line, which is easy to deal with problems concerning general variable coefficients. Some composite generalized Laguerre–Legendre interpolations on the whole line are established. The convergence and stability of the proposed algorithm are proved. Numerical results show the efficiency of this approach and agree well with the theoretical analysis.

A multi-domain Galerkin method with numerical integration for the Burgers equation