Splitting Schemes for Problems with Memory

Abstract The paper considers the Cauchy problem for a first-order integro-differential equation with memory in a finite-dimensional Hilbert space. The main computational difficulty of such problems is the need to store and process the solution at all previous time levels. To overcome this difficulty, an approach is used that approximates the memory kernel by a sum of exponentials, which reduces the original nonlocal problem to a local one – a system of weakly coupled evolution equations with additional ordinary differential equations for auxiliary functions. The problem is formulated in vector form on the direct sum of Hilbert spaces. Unconditional stability of two-level operator-difference schemes with weights is proved under standard restrictions. Splitting schemes are proposed and investigated by separating the local and integral operators of the problem. Possibilities for constructing similar schemes for other nonlocal problems, in particular for the equation with memory of the time derivative of the solution, are noted.