In this work, high order splitting methods have been used for calculating the numerical solutions of the Burgers' equation in one space dimension with periodic and Dirichlet boundary conditions. However, splitting methods with real coefficients of order higher than two necessarily have negative coefficients and can not be used for time-irreversible systems, such as Burgers equations, due to the time-irreversibility of the Laplacian operator. Therefore, the splitting methods with complex coefficients and extrapolation methods with real and positive coefficients have been employed. If we consider the system as the perturbation of an exactly solvable problem(or can be easily approximated numerically), it is possible to employ highly efficient methods to approximate Burgers' equation. The numerical results show that the methods with complex time steps having one set of coefficients real and positive, say a i ∈ R + and b i ∈ C + , and high order extrapolation methods derived from a lower order splitting method produce very accurate solutions of the Burgers' equation.
We consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative coefficients). We propose to consider a class of methods that allows us to evaluate all timedependent operators at real values of the time, leading to schemes which are stable and simple to implement. If the system can be considered as the perturbation of an exactly solvable problem and the flow of the dominant part is advanced using real coefficients, it is possible to build highly efficient methods for these problems. We show the performance of this class of methods on several numerical examples and present some new improved schemes.
We propose splitting methods for the computation of the exponential of perturbed matrices which can be written as the sum A = D+εB of a sparse and efficiently exponentiable matrix D with sparse exponential e D and a dense matrix εB which is of small norm in comparison with D. The predominant algorithm is based on scaling the large matrix A by a small number 2 −s , which is then exponentiated by efficient Padé or Taylor methods and finally squared in order to obtain an approximation for the full exponential. In this setting, the main portion of the computational cost arises from dense-matrix multiplications and we present a modified squaring which takes advantage of the smallness of the perturbed matrix B in order to reduce the number of squarings necessary. Theoretical results on local error and error propagation for splitting methods are complemented with numerical experiments and show a clear improvement over existing methods when medium precision is sought.
SCALING, SPLITTING, AND SQUARING
595In some cases, if the matrix A has a given structure, more efficient methods can be obtained [4,5] . For example, to compute the exponential of upper or lower triangular matrices, in [1], the authors show that it is advantageous to exploit the fact that the diagonal elements of the exponential are exactly known. It is then more efficient to replace the diagonal elements obtained using, e.g., Taylor or Padé approximations by the exact solution before squaring the matrix (this technique can also be extended to the first super (or sub-)diagonal elements). On the other hand, in many cases the matrix A can be considered as a small perturbation of a sparse matrix D, i.e., A = D + B with B < D (and frequently B D ), where e D is sparse and exactly solvable (or can be accurately and cheaply approximated numerically), and B is a dense matrix. This is the case, for example, if D is diagonal (or block diagonal with small matrices along the diagonal), or if it is diagonalizable using only a few elementary transforms. Another example with similar structure is given by
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