Splitting schemes for problems with memory

Technical Summary: Splitting Schemes for Problems with Memory

Problem Statement
The paper addresses the numerical solution of the Cauchy problem for first-order integro-differential equations with memory in a finite-dimensional Hilbert space. The governing equation is of the form:
$$\frac{du}{dt} + Au + \int_0^t k(t-s)Bu(s)ds = f(t)$$
where $A$ and $B$ are operators, and $k(t)$ is a memory kernel. The primary computational challenge identified is the nonlocal nature of the integral term, which necessitates storing and processing the solution at all previous time levels. This requirement leads to a significant increase in computational costs and memory usage as the simulation progresses.

Methodology
To overcome the computational burden of nonlocality, the author employs a localization strategy. The memory kernel $k(t)$ is approximated by a sum of exponential functions:
$$k(t) = \sum_{i=1}^m a_i \exp(-b_i t)$$
This approximation transforms the original integro-differential equation into a system of local, weakly coupled evolution equations. By introducing auxiliary functions $u_i(t)$ defined as convolutions of the solution with the exponential terms, the nonlocal problem is reduced to a system of ordinary differential equations (ODEs) coupled with the main evolution equation.

The problem is then reformulated in a vector form on the direct sum of Hilbert spaces. The resulting system is treated as a first-order evolution equation with a composite operator structure. The paper investigates two-level operator-difference schemes with weights ($\sigma$) for time discretization.

The core methodological contribution involves the application of splitting schemes (additive operator-difference schemes). The operator of the system is decomposed additively to separate the local differential operators from the integral (memory) operators, or to decompose the spatial operators $A$ and $B$ into simpler components. The paper specifically analyzes:

1. Component-wise splitting: Solving subproblems sequentially for individual operator terms.
2. Decomposition of operators: Separating the main operator $A$ (modeling local processes) from the operator $B$ (modeling memory effects) and further decomposing them based on additive representations $A = \sum A_\alpha$ and $B = \sum B_\beta$.

Key Contributions and Results

1. Localization and Stability: The paper demonstrates that approximating the kernel by a sum of exponentials allows the nonlocal problem to be converted into a local system. Theorem 1 establishes an a priori stability estimate for the continuous extended system, showing that the norm of the solution (including auxiliary functions) is bounded by the initial data.
2. Unconditional Stability of Discrete Schemes:
   • For the weighted two-level scheme applied to the vector problem, Theorem 2 proves unconditional stability in the Hilbert space $H_B$ provided the weight parameter $\sigma \geq 0.5$.
   • Theorem 3 extends this result to component-wise splitting schemes, proving that the additive decomposition of the operator $A$ preserves unconditional stability under the same weight restriction ($\sigma \geq 0.5$).
3. Splitting Strategies:
   • The paper proposes a specific splitting strategy where the local operator $A$ and the memory-related operator terms are separated. This allows solving simpler subproblems (e.g., solving for the main variable without memory in one step, and updating auxiliary variables in another) rather than inverting a large coupled system at every time step.
   • The author notes that while splitting the operator $A$ is straightforward, splitting the operator $B$ (associated with the time derivative in the integral term) is more complex and requires specific factorization conditions (e.g., $B = C^*C$) to construct stable additive schemes.
4. Extension to Derivative Memory: The methodology is extended to a different class of problems where the memory term acts on the time derivative of the solution:
   $$\frac{du}{dt} + Au + \int_0^t k(t-s)B \frac{du}{ds}(s) ds = 0$$
   The paper formulates the corresponding vector problem and proves unconditional stability for symmetric schemes ($\sigma = 0.5$) and general weights ($\sigma \geq 0.5$) under non-negativity conditions for the operators.

Significance and Claims
The paper claims that the proposed approach effectively reduces the computational complexity of solving nonlocal evolution equations. By transforming the problem into a local system of weakly coupled equations, the need to store the entire history of the solution is eliminated, replaced by the storage of a fixed number of auxiliary variables.

The significance lies in the construction of unconditionally stable splitting schemes that simplify the computational work at each new time level. Instead of solving a complex coupled system involving both local and nonlocal operators simultaneously, the proposed schemes allow for the sequential solution of simpler subproblems. The author explicitly states that while decomposing the main operator $A$ offers substantial simplification, the possibilities for decomposing the additional operator associated with the memory term are more limited, though feasible under specific structural assumptions.

The work is presented as a theoretical investigation into the stability and construction of these schemes, with the goal of facilitating the computational implementation of implicit schemes for non-stationary problems with memory. The results are framed as providing a rigorous foundation for using additive-operator difference schemes in this context, ensuring stability without restrictive time-step limitations.