arXiv subjects

Zhenning Cai

Publications and source records attributed to Zhenning Cai.

At least 19 recordsLinked to original sources

arXiv2026

A General First- and Second-Order Numerical Solver for Non-Markovian Quantum State Diffusion

The numerical simulation of non-Markovian open quantum systems based on the non-Markovian quantum state diffusion (NMQSD) equation is complicated by functional derivatives with respect to the stochastic process. A general numerical framework that directly treats these functional derivatives without relying on prescribed decompositions of the bath correlation function is still lacking. In this work, we derive an analytical solution of the linear NMQSD equation that reveals three elementary structures of the non-Markovian stochastic dynamics: stochastic propagation, functional-derivative insertion, and memory pairing. Based on this structure, we construct a general auxiliary-state framework for arbitrary bath correlation functions. The framework separates the numerical construction into time discretization, memory quadrature, and hierarchy truncation. We then construct first- and second-order schemes and provide diagrammatic transition rules for their explicit implementation. Numerical results verify the expected temporal accuracy and demonstrate the applicability of the proposed methods to different bath correlation functions and multi-level quantum systems.

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arXiv2026

A low-rank hierarchical framework for the non-Markovian stochastic Schr\"odinger equation with convergence analysis

We propose and analyze a novel numerical framework for the non-Markovian stochastic Schr\"odinger equation (NMSSE) based on a low-rank approximation of the bath correlation functions. By decomposing the memory kernel into a finite-dimensional representation, we derive a truncated system of hierarchical equations that effectively balances computational tractability with physical fidelity. A rigorous convergence analysis is established for the hierarchical framework under mild assumptions. We demonstrate that our formulation serves as a mathematical generalization of the Hierarchy of Pure States (HOPS), encompassing it as a special case while offering a more flexible representation of non-Markovian effects. Numerical experiments across several benchmark models are presented to illustrate the validity and efficacy of the proposed method.

math.NA↗

arXiv2026

Accelerated iterative method for solving the steady-state Boltzmann equation

The efficient simulation of steady-state rarefied gas flows remains a significant computational challenge due to the high dimensionality of the collision integral and the severe numerical stiffness in the near-continuum regime. In this work, we propose a modified Newton method equipped with a macroscopic synthetic system (Newton-MS) for the steady-state Boltzmann equation with the quadratic collision operator. In Newton-MS, the modified Newton iteration is utilized as the outer nonlinear solver, while each Newton correction equation is solved by an inner source iteration, where the linearized collision operator is utilized to approximate the quadratic collision model, and it is reduced into a linear iteration. Moreover, a macroscopic synthetic system based on Chapman-Enskog closure is derived to accelerate the convergence of the linear inner iteration in the continuum limit. Besides, the fully discrete macroscopic synthetic system is deduced under the framework of the discontinuous Galerkin method to reduce computational cost compared to directly discretizing the continuous macroscopic synthetic system. Several numerical examples, including the 1D Fourier, Couette flow problem, and the 2D cavity flow and thermal-driven cavity flow, are studied to validate the high efficiency of Newton-MS.

math.NA↗

arXiv2026

Global well-posedness of the linearized R13 moment equations with Onsager boundary conditions

This paper establishes the global well-posedness of the linearized regularized 13-moment (R13) equations for rarefied gas flows. We first derive an entropy inequality for the system on bounded domains subject to Onsager boundary conditions. For the steady-state problem, well-posedness is proved via the Ladyzhenskaya-Babuska-Brezzi (LBB) theorem, facilitated by novel boundary-related Korn-type inequalities. Furthermore, leveraging the Lumer-Phillips theorem, we extend these results to guarantee the global well-posedness of the time-dependent R13 equations. Our theoretical framework uniformly accommodates the models for both Maxwell and general non-Maxwell molecules.

math.AP↗

arXiv2026

Asymptotic Analysis of Shallow Water Moment Equations

The Shallow Water Moment Equations (SWME) are an extension of the Shallow Water Equations (SWE) for improved modelling of free-surface flows. In contrast to the SWE, the SWME incorporate vertical velocity profile information. The SWME framework approximates vertical velocity profiles using a polynomial expansion with Legendre polynomials and polynomial coefficients, also called moment variables. The SWME have an increased number of variables that must always be incorporated, even when the flow approaches a viscous slip equilibrium state that could be characterised by vanishing moment variables. To reduce the complexity of the SWME in cases proximate to this equilibrium, we conduct an asymptotic analysis of the SWME. This yields the closed form Reduced Shallow Water Moment Equations (RSWME) for deviations from the equilibrium. The RSWME have fewer variables, compared to the SWME. The hyperbolicity of the RSWME is analysed. Numerical tests include a wave with a sharp height gradient, a smoother height gradient and a square root velocity profile. The numerical tests demonstrate that the RSWME reduce computational cost up to 77% compared to the SWME and improves accuracy up to 88% over the SWE.

math.NA↗

arXiv2026

Stability and Convergence of Mixed Finite Elements for Linear Regularized 13-Moment Equations

We present a stable and convergent mixed finite element method (MFEM) for the linear regularized 13-moment (R13) equations in rarefied gas dynamics. Unlike existing methods that require stabilization via penalty terms, our scheme achieves inherent stability by enriching the finite element basis with bubble functions. We provide a rigorous theoretical analysis, establishing second-order convergence rates in the $L^2$ norm under mild regularity assumptions. Beyond theoretical properties, our scheme demonstrates practical advantages over standard MFEM schemes, yielding robust numerical results even in the presence of geometric singularities.

math.NA↗

arXiv2025

Second-order discretization of Dyson series: iterative method, numerical analysis and applications in open quantum systems

We propose a general strategy to discretize the Dyson series without applying direct numerical quadrature to high-dimensional integrals, and extend this framework to open quantum systems. The resulting discretization can also be interpreted as a Strang splitting combined with a Taylor expansion. Based on this formulation, we develop a numerically exact iterative method for simulation system-bath dynamics. We propose two numerical schemes, which are first-order and second-order in time step $\Delta t$ respectively. We perform a rigorous numerical analysis to establish the convergence orders of both schemes, proving that the global error decreases as $\mathcal{O}(\Delta t)$ and $\mathcal{O}(\Delta t^2)$ for the first- and second-order methods, respectively. In the second-order scheme, we can safely omitted most terms arising from the Strang splitting and Taylor expansion while maintaining second-order accuracy, leading to a substantial reduction in computational complexity. For the second-order method, we achieves a time complexity of $\mathcal{O}(M^3 2^{2K_{\max}} K_{\max}^2)$ and a space complexity of $\mathcal{O}(M^2 2^{2K_{\max}} K_{\max})$ where $M$ denotes the number of system levels and $K_{\max}$ the number of time steps within the memory length. Compared with existing methods, our approach requires substantially less memory and computational effort for multilevel systems ($M\geqslant 3$). Numerical experiments are carried out to illustrate the validity and efficiency of our method.

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arXiv2025

Reducing Spatial and Temporal Dimensionality in the Multidimensional Caldeira-Leggett Model

Focusing on the real-time dynamics of the reduced density matrix of the multidimensional Caldeira-Leggett model, several techniques are adopted in this paper to reduce the spatial and temporal dimensionality, combined into an efficient algorithm. From a spatial perspective, an equivalent formulation of the Dyson series is presented. With the aid of a low-rank approximation, the spatial dimensionality of open quantum system simulations is halved. From a temporal perspective, the frozen Gaussian approximation is used to approximate both the evolution operator and the interaction operator in the multidimensional Caldeira-Leggett model. This reduces the high-dimensional integrals to one- and two-dimensional integrals independent of the truncation level of the Dyson series. Through these techniques, we design an efficient algorithm whose validity is verified through several numerical experiments, including a two-dimensional double slit simulation.

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arXiv2025

Accelerated Inchworm Method with Tensor-Train Bath Influence Functional

We propose an efficient tensor-train-based algorithm for simulating open quantum systems with the inchworm method, where the reduced dynamics of the open quantum system is expressed as a perturbative series of high-dimensional integrals. Instead of evaluating the integrals with Monte Carlo methods, we approximate the costly bath influence functional (BIF) in the integrand as a tensor train, allowing accurate deterministic numerical quadrature schemes implemented in an iterative manner. Thanks to the low-rank structure of the tensor train, our proposed method has a complexity that scales linearly with the number of dimensions. Our method couples seamlessly with the tensor transfer method, allowing long-time simulations of the dynamics.

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arXiv2025

A Framework of Model Reduction with Arbitrary Orders of Accuracy for the Boltzmann Equation

This paper presents a general framework for constructing reduced models that approximate the Boltzmann equation with arbitrary orders of accuracy in terms of the Knudsen number $\mathit{Kn}$, applicable to general collision models in rarefied gas dynamics. The framework is based on an orthogonal decomposition of the distribution function into components of different orders in $\mathit{Kn}$, from which the reduced models are systematically derived through asymptotic analysis. Compared to the Chapman-Enskog expansion, our approach yields more tractable model structures. Notably, we establish that a reduced model retaining all terms up to $O(\mathit{Kn}^n)$ in the expansion surprisingly yields models with order of accuracy $O(\mathit{Kn}^{n+1})$. Furthermore, when the collision term is linearized, the accuracy improves dramatically to $O(\mathit{Kn}^{2n})$. These results extend to regularized models containing second-order derivatives. As concrete applications, we explicitly derive 13-moment systems of Burnett and super-Burnett orders valid for arbitrary collision models.

math-ph↗

arXiv2025

A fast Fourier spectral method for the linearized Boltzmann collision operator

We introduce a fast Fourier spectral method to compute linearized collision operators of the Boltzmann equation for variable hard-sphere gases. While the state-of-the-art method provides a computational cost O(MN^4 log N), with N being the number of modes in each direction and M being the number of quadrature points on a hemisphere, our method reduces the cost to O(N^4 log N), removing the factor M, which could be large in our numerical tests. The method is applied in a numerical solver for the steady-state Boltzmann equation with quadratic collision operators. Numerical experiments for both spatially homogeneous and inhomogeneous Boltzmann equations have been carried out to test the accuracy and efficiency of our method.

math.NA↗

arXiv2024

Comment on "Unified framework for open quantum dynamics with memory"

A recent article by Ivander, Lindoy and Lee [Nature Communications 15, 8087 (2024)] claims to discover the relationship between the generalized quantum master equation (GQME) and the path integral for a system coupled to a harmonic bath. However, this relationship was already established in 2020 by Makri in the context of the small matrix decomposition of the path integral (SMatPI) [J. Chem. Theory and Comput. 16, 4038 (2020)]. The procedure that this article uses in its Supplementary Information (SI) to obtain the various matrices follows the SMatPI decomposition steps for the alternative Trotter ordering. The absence of endpoint effects in the kernel matrices of the discretized GQME expression for the reduced density matrix (RDM) is the consequence of a crude GQME discretization and is not consistent with the SMatPI decomposition of an auxiliary matrix presented in the SI. This form is identical to the transfer tensor method (TTM) of Cerrillo and Cao [Phys. Rev. Lett. 112, 110401 (2014)]. Further, the Dyck path section of this article follows precisely the diagrammatic analysis developed by Wang and Cai in a recent paper [Communications in Computational Physics 36, 389 (2024)]. We elaborate on these three critiques in this Comment.

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arXiv2024

Symmetric Gauss-Seidel Method with a Preconditioned Fixed-Point Iteration for the Steady-State Boltzmann equation

We introduce a numerical solver for the steady-state Boltzmann equation based on the symmetric Gauss-Seidel (SGS) method. To solve the nonlinear system on each grid cell derived from the SGS method, a fixed-point iteration preconditioned with its asymptotic limit is developed. The preconditioner only requires solving an algebraic system which is easy to implement and can speed up the convergence significantly especially in the case of small Knudsen numbers. Additionally, we couple our numerical scheme with the multigrid method to accelerate convergence. A variety of numerical experiments are carried out to illustrate the effectiveness of these methods.

math.NA↗

arXiv2024

Solving Caldeira-Leggett Model by Inchworm Method with Frozen Gaussian Approximation

We propose an algorithm that combines the inchworm method and the frozen Gaussian approximation to simulate the Caldeira-Leggett model in which a quantum particle is coupled with thermal harmonic baths. In particular, we are interested in the real-time dynamics of the reduced density operator. In our algorithm, we use frozen Gaussian approximation to approximate the wave function as a wave packet in integral form. The desired reduced density operator is then written as a Dyson series, which is the series expression of path integrals in quantum mechanics of interacting systems. To compute the Dyson series, we further approximate each term in the series using Gaussian wave packets, and then employ the idea of the inchworm method to accelerate the convergence of the series. The inchworm method formulates the series as an integro-differential equation of ``full propagators'', and rewrites the infinite series on the right-hand side using these full propagators, so that the number of terms in the sum can be significantly reduced, and faster convergence can be achieved. The performance of our algorithm is verified numerically by various experiments.

quant-ph↗

arXiv2024

Iterative methods of linearized moment equations for rarefied gases

We study the iterative methods for large moment systems derived from the linearized Boltzmann equation. By Fourier analysis, it is shown that the direct application of the block symmetric Gauss-Seidel (BSGS) method has slower convergence for smaller Knudsen numbers. Better convergence rates for dense flows are then achieved by coupling the BSGS method with the micro-macro decomposition, which treats the moment equations as a coupled system with a microscopic part and a macroscopic part. Since the macroscopic part contains only a small number of equations, it can be solved accurately during the iteration with a relatively small computational cost, which accelerates the overall iteration. The method is further generalized to the multiscale decomposition which splits the moment system into many subsystems with different orders of magnitude. Both one- and two-dimensional numerical tests are carried out to examine the performances of these methods. Possible issues regarding the efficiency and convergence are discussed in the conclusion.

math.NA↗

arXiv2024

Time-dependent Regularized 13-Moment Equations with Onsager Boundary Conditions in the Linear Regime

We develop the time-dependent regularized 13-moment equations for general elastic collision models under the linear regime. Detailed derivation shows the proposed equations have super-Burnett order for small Knudsen numbers, and the moment equations enjoy a symmetric structure. A new modification of Onsager boundary conditions is proposed to ensure stability as well as the removal of undesired boundary layers. Numerical examples of one-dimensional channel flows is conducted to verified our model.

math.AP↗

arXiv2024

Simulation of Spin Chains with off-diagonal Coupling Using Inchworm Method

We study the dynamical simulation of open quantum spin chain with nearest neighboring coupling, where each spin in the chain is associated with a harmonic bath. This is an extension of our previous work [G. Wang and Z. Cai, J. Chem. Theory Comput., 19, 8523--8540, 2023] by generalizing the application of the inchworm method and the technique of modular path integrals from diagonally coupled cases to off-diagonally coupled cases. Additionally, to reduce computational and memory cost in long time simulation, we apply tensor-train representation to efficiently represent the reduced density matrix of the spin chains, and employ the transfer tensor method (TTM) to avoid exponential growth of computational cost with respect to time. Abundant numerical experiments are performed to validate our method.

quant-ph↗

arXiv2023

Tree-based Implementation of the Small Matrix Path Integral for System-Bath Dynamics

The small matrix path integral (SMatPI) method is an efficient numerical approach to simulate the evolution of a quantum system coupled to a harmonic bath. The method relies on a sequence of kernel matrices that defines the non-Markovian dynamics of the quantum system. In the original SMatPI method, these kernels are computed indirectly through the QuAPI method. Instead, we focus on the definition of the kernel matrices and reveal a recurrence relation in these matrices. Using such a relationship, a tree based algorithm (t-SMatPI) is developed, which is shown to be much faster than straightforward computation of the kernel matrices based on their definitions. This algorithm bypasses the step to compute the SMatPI matrices by other path integral methods and provides more understanding of the SMatPI matrices themselves. Meanwhile, it keeps the memory cost and computational cost low. Numerical experiments show that the t-SMatPI algorithm gives exactly the same result as i-QuAPI and SMatPI. In spite of this, our method may indicate some new properties of open quantum systems, and has the potential to be generalized to higher-order numerical schemes.

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