Optimized high-order splitting methods for some classes of parabolic equations (Q2840621)

The subject of this paper is an application of the splitting method to evolution differential equations of the following form NEWLINE\[NEWLINE{du\over dt}(t)=Au(t)+Bu(t),\;\;u(0)=u_0, \leqno(1)NEWLINE\]NEWLINE where \(A\), \(B\) are operators defined over a Banach space \(X\). The primitive form of the splitting method consists in an approximation of the solution of the equation (1) by the solution of two independent equations NEWLINE\[NEWLINE{dv\over dt}(t)=Av(t)\;\;\text{and}\;\;{dw\over dt}(t)=Bw(t)\leqno (2)NEWLINE\]NEWLINE with properly chosen initial values, on a time interval \(t\in [0,h]\) with small \(h>0\). Splitting is an important tool for numerical methods, and also for the analysis of solutions of (1) in various applications. The operation of splitting is feasible under additional conditions. For example, if \(A\), \(B\), and \(A+B\) generate \({\mathcal C}^0\), a semigroup for \(t\geq 0\) over the space \(X\), then splitting is possible.NEWLINENEWLINEThe splitting method is well known from the early sixties of the twentieth century, and there exists a rich literature of this subject. Under certain conditions the method of the form (2) can reach the degree of approximation \(O(h)\) or \(O(h^2)\). This work is devoted to the question how to build the method of higher order of approximation. In fact, it is possible to make even a very high-order splitting, taking methods which may be written in a `semi-group language' as NEWLINE\[NEWLINEe^{ha_1A}e^{hb_1B}e^{ha_2A}e^{hb_2B}\cdots.NEWLINE\]NEWLINE The authors discuss the problem of the choice of the coefficients \(a_i\), \(b_i\). Because of the semigroup \({\mathcal C}^0\) properties, real positive coefficients are admitted. But the authors take into account also complex coefficients with positive real parts, and find new splittings of higher order. The paper contains certain propositions of sets of coefficients for a splitting of order 6 and 8. A rich bibliography is joined.