Related papers: Multi-product operator splitting as a general meth…
The splitting of $\e^{h(A+B)}$ into a single product of $\e^{h A}$ and $\e^{hB}$ results in symplectic integrators when $A$ and $B$ are classical Lie operators. However, at high orders, a single product splitting, with exponentially growing…
In approximating solutions of nonstationary problems, various approaches are used to compute the solution at a new time level from a number of simpler (sub-)problems. Among these approaches are splitting methods. Standard splitting schemes…
Dissipation and irreversibility are central to most physical processes, yet they lead to non-unitary dynamics that are challenging to realise on quantum processors. High-order operator splitting is an attractive approach for simulating…
Symplectic integration methods based on operator splitting are well established in many branches of science. For Hamiltonian systems which split in more than two parts, symplectic methods of higher order have been studied in detail only for…
Operator splitting methods allow to split the operator describing a complex dynamical system into a sequence of simpler subsystems and treat each part independently. In the modeling of dynamical problems, systems of (possibly coupled)…
We provide general product formulas for the solutions of non-autonomous abstract Cauchy problems. The main technical tool is the application of evolution semigroup methods, allowing the direct application of existing results on autonomous…
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…
By exploiting the error functions of explicit symplectic integrators for solving separable Hamiltonians, I show that it is possible to develop explicit, time-reversible symplectic integrators for solving non-separable Hamiltonians of the…
The explicit split-operator algorithm is often used for solving the linear and nonlinear time-dependent Schr\"{o}dinger equations. However, when applied to certain nonlinear time-dependent Schr\"{o}dinger equations, this algorithm loses…
Splitting methods have emerged as powerful tools to address complex problems by decomposing them into smaller solvable components. In this work, we develop a general approach to forward-backward splitting methods for solving monotone…
We propose a high order numerical decomposition of exponentials of hermitean operators in terms of a product of exponentials of simple terms, following an idea which has been pioneered by M. Suzuki, however implementing it for complex…
Operator splitting methods tailored to coupled linear port-Hamiltonian systems are developed. We present algorithms that are able to exploit scalar coupling, as well as multirate potential of these coupled systems. The obtained algorithms…
We provide a general product formula for the solution of nonautonomous abstract delay equations. After having shown the convergence we obtain estimates on the order of convergence for differentiable history functions. Finally, the…
In this paper, we develop high-order splitting methods for linear port-Hamiltonian systems, focusing on preserving their intrinsic structure, particularly the dissipation inequality. Port-Hamiltonian systems are characterized by their…
By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on $\R^n$ as an infinite product of zero-order pseudodifferential operators. A similar representation formula is…
The operator splitting method has been widely used to solve differential equations by splitting the equation into more manageable parts. In this work, we resolves a long-standing problem -- how to establish the stability of multi-product…
The solution of many physical evolution equations can be expressed as an exponential of two or more operators acting on initial data. Accurate solutions can be systematically derived by decomposing the exponential in a product form. For…
Nonlinear elliptic problems arise in many fields, including plasma physics, astrophysics, and optimal transport. In this article, we propose a novel operator-splitting/finite element method for solving such problems. We begin by introducing…
Operator-splitting methods are widely used to solve differential equations, especially those that arise from multi-scale or multi-physics models, because a monolithic (single-method) approach may be inefficient or even infeasible. The most…
In this paper, we combine the operator splitting methodology for abstract evolution equations with that of stochastic methods for large-scale optimization problems. The combination results in a randomized splitting scheme, which in a given…