Step-by-step convolution

Propagation of elastic waves in multiphase media “with memory” is described by the system of the partial integrodifferential equations. Integral terms of these equations are convolutions of the computed solution in all time moments previous to time moment under consideration with known kernels depending on properties of the studied medium. Construction of economic finite difference schemes for calculation of wave fields are confronted with difficulty of convolution calculation, since quantity of arithmetic operations for calculation of one value of convolution is proportional to number of steps on time. In present work, the way of overcoming of arising difficulty is offered. Continuous recurrent algorithms are considered for this purpose. In particular, linear algorithms lead to a method of kernel approximation by the sum of exponents, and for any continuous algorithms the lower estimate is proved to their accuracy depending on volume of memory used by them. On a concrete example of a power-like kernel it is shown, that the linear algorithm is one of the best among all continuous algorithms.