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This doctoral thesis provides a comprehensive numerical analysis and exploration of several stochastic partial differential equations (SPDEs). More specifically, this thesis investigates time integrators for SPDEs with white noise dispersion. The thesis begins by examining the stochastic nonlinear Schrödinger equation with white noise dispersion (SNLSE), see Paper 1. The investigation probes the performance of different numerical integrators for this equation, focusing on their convergences, L2-norm preservation, and computational efficiency. Further, this thesis thoroughly investigates a conjecture on the critical exponent of the SNLSE, related to a phenomenon known as blowup, through numerical means. The thesis then introduces and studies exponential integrators for the stochastic Manakov equation (SME) by presenting two new time integrators - the explicit and symmetric exponential integrators - and analyzing their convergence properties, see Paper 2. Notably, this study highlights the flexibility and efficiency of these integrators compared to traditional schemes. The narrative then turns to the Lie-Trotter splitting integrator for the SME, see Paper 3, comparing its performance to existing time integrators. Theoretical proofs for convergence in various senses, alongside extensive numerical experiments, shed light on the efficacy of the proposed numerical scheme. The thesis also deep dives into the critical exponents of the SME, proposing a conjecture regarding blowup conditions for this SPDE. Lastly, the focus shifts to the stochastic generalized Benjamin-Bona-Mahony equation, see Paper 4. The study introduces and numerically assesses four novel exponential integrators for this equation. A primary finding here is the superior performance of the symmetric exponential integrator. This thesis also offers a succinct and novel method to depict the order of convergence in probability.

This doctoral thesis provides a comprehensive numerical analysis and exploration of several stochastic partial differential equations (SPDEs). More specifically, this thesis investigates time integrators for SPDEs with white noise dispersion. The thesis begins by examining the stochastic nonlinear Schrödinger equation with white noise dispersion (SNLSE), see Paper 1. The investigation probes the performance of different numerical integrators for this equation, focusing on their convergences, L2-norm preservation, and computational efficiency. Further, this thesis thoroughly investigates a conjecture on the critical exponent of the SNLSE, related to a phenomenon known as blowup, through numerical means. The thesis then introduces and studies exponential integrators for the stochastic Manakov equation (SME) by presenting two new time integrators - the explicit and symmetric exponential integrators - and analyzing their convergence properties, see Paper 2. Notably, this study highlights the flexibility and efficiency of these integrators compared to traditional schemes. The narrative then turns to the Lie-Trotter splitting integrator for the SME, see Paper 3, comparing its performance to existing time integrators. Theoretical proofs for convergence in various senses, alongside extensive numerical experiments, shed light on the efficacy of the proposed numerical scheme. The thesis also deep dives into the critical exponents of the SME, proposing a conjecture regarding blowup conditions for this SPDE. Lastly, the focus shifts to the stochastic generalized Benjamin-Bona-Mahony equation, see Paper 4. The study introduces and numerically assesses four novel exponential integrators for this equation. A primary finding here is the superior performance of the symmetric exponential integrator. This thesis also offers a succinct and novel method to depict the order of convergence in probability.

This doctoral thesis provides a comprehensive numerical analysis and exploration of several stochastic partial differential equations (SPDEs). More specifically, this thesis investigates time integrators for SPDEs with white noise dispersion. The thesis begins by examining the stochastic nonlinear Schrödinger equation with white noise dispersion (SNLSE), see Paper 1. The investigation probes the performance of different numerical integrators for this equation, focusing on their convergences, L2-norm preservation, and computational efficiency. Further, this thesis thoroughly investigates a conjecture on the critical exponent of the SNLSE, related to a phenomenon known as blowup, through numerical means. The thesis then introduces and studies exponential integrators for the stochastic Manakov equation (SME) by presenting two new time integrators - the explicit and symmetric exponential integrators - and analyzing their convergence properties, see Paper 2. Notably, this study highlights the flexibility and efficiency of these integrators compared to traditional schemes. The narrative then turns to the Lie-Trotter splitting integrator for the SME, see Paper 3, comparing its performance to existing time integrators. Theoretical proofs for convergence in various senses, alongside extensive numerical experiments, shed light on the efficacy of the proposed numerical scheme. The thesis also deep dives into the critical exponents of the SME, proposing a conjecture regarding blowup conditions for this SPDE. Lastly, the focus shifts to the stochastic generalized Benjamin-Bona-Mahony equation, see Paper 4. The study introduces and numerically assesses four novel exponential integrators for this equation. A primary finding here is the superior performance of the symmetric exponential integrator. This thesis also offers a succinct and novel method to depict the order of convergence in probability.

This doctoral thesis provides a comprehensive numerical analysis and exploration of several stochastic partial differential equations (SPDEs). More specifically, this thesis investigates time integrators for SPDEs with white noise dispersion. The thesis begins by examining the stochastic nonlinear Schrödinger equation with white noise dispersion (SNLSE), see Paper 1. The investigation probes the performance of different numerical integrators for this equation, focusing on their convergences, L2-norm preservation, and computational efficiency. Further, this thesis thoroughly investigates a conjecture on the critical exponent of the SNLSE, related to a phenomenon known as blowup, through numerical means. The thesis then introduces and studies exponential integrators for the stochastic Manakov equation (SME) by presenting two new time integrators - the explicit and symmetric exponential integrators - and analyzing their convergence properties, see Paper 2. Notably, this study highlights the flexibility and efficiency of these integrators compared to traditional schemes. The narrative then turns to the Lie-Trotter splitting integrator for the SME, see Paper 3, comparing its performance to existing time integrators. Theoretical proofs for convergence in various senses, alongside extensive numerical experiments, shed light on the efficacy of the proposed numerical scheme. The thesis also deep dives into the critical exponents of the SME, proposing a conjecture regarding blowup conditions for this SPDE. Lastly, the focus shifts to the stochastic generalized Benjamin-Bona-Mahony equation, see Paper 4. The study introduces and numerically assesses four novel exponential integrators for this equation. A primary finding here is the superior performance of the symmetric exponential integrator. This thesis also offers a succinct and novel method to depict the order of convergence in probability.

Magnetophoresis is one of the most popular particle manipulation schemes used in microfluidic devices. Predicting the performance of arbitrary magnetophoretic module via simulation is crucial for the realization of high-performance, purpose-built magnetophoretic devices. However, the simulation process may take a long time, slowing down the simulation throughput. This work presents a novel algorithm to rapidly predict the trajectory of magnetic particles in microfluidic devices with arbitrary magnetophoretic structures under continuous fluidic flow conditions. Based on the Eulerian–Lagrangian approach, by employing numerically calculated magnetic force and fluidic velocity datasets obtained by finite element analysis software, we used an exponential integrator inspired numerical method to solve for the motion of magnetic particles in the system with the assumption that the magnetic and fluidic forces are constants within a small period of time. The results showed that, for our application, the order of truncation error of our algorithm is comparable with the 4th Order Runge–Kutta method and commercial Runge–Kutta based solver ODE45 from MATLAB. Furthermore, our algorithm is inherently stable, thus enabling the use of arbitrary time stepping values without the need of adaptive step size adjustment, which significantly reduces simulation time when compared to the 4th Order Runge–Kutta method and ODE45. In our simulations, a maximum of 40 times reduction in simulation time can be achieved. The proposed algorithm and the performed simulations are validated experimentally. © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

The growing proportion of distributed generators in active distribution networks highlights the importance of multi-timescale system dynamics. The electromagnetic transient simulators with detailed modeling have been widely used for high-fidelity simulations, which leads to a contradiction between the efficiency and accuracy in the studies of the multi-timescale property of the ADNs. The shifted-frequency analysis method integrated with state-space based exponential integration is introduced in this paper, which possesses both high accuracy of exponential integration and the advantage of system-level modeling. It accurately simulates the networks considering the nonlinearities with the advantage that allows large time-steps in the simulations wherein the signals have a small bandwidth around the fundamental frequency. A model-switching method based on the integral transformation of the shifted-frequency models is designed to address the inefficiency of multi-timescale simulations. In addition, a flexible discontinuity treatment in the shifted-frequency domain is introduced for the reduction of the errors brought by the detections and interpolations of discontinuities in the time-domain. Numerical studies are conducted considering the distributed generations, the shifted-frequency models are generated integrally from the original time-domain models. The results of the multi-timescale simulations with model-switching in different domains show the high efficiency of the proposed method, and the discontinuity treatment presents higher accuracy compared with the time-domain solution. Those combined advantages compose the flexible shifted-frequency analysis for the multi-timescale simulations of active distribution networks.

This article presents explicit exponential integrators for stochastic Maxwell's equations driven by both multiplicative and additive noises. By utilizing the regularity estimate of the mild solution, we first prove that the strong order of the numerical approximation is ½ for general multiplicative noise. Combining a proper decomposition with the stochastic Fubini's theorem, the strong order of the proposed scheme is shown to be 1 for additive noise. Moreover, for linear stochastic Maxwell's equation with additive noise, the proposed time integrator is shown to preserve exactly the symplectic structure, the evolution of the energy as well as the evolution of the divergence in the sense of expectation. Several numerical experiments are presented in order to verify our theoretical findings.

type:Journal Article, We propose new explicit exponential Runge-Kutta methods for the weak approximation of solutions of stiff Itô stochastic differential equations (SDEs). We also consider the use of exponential Runge-Kutta methods in combination with splitting methods. These methods have weak order 2 for multidimensional, noncommutative SDEs with a semilinear drift term, whereas they are of order 2 or 3 for semilinear ordinary differential equations. These methods are A-stable in the mean square sense for a scalar linear test equation whose drift and diffusion terms have complex coefficients. We carry out numerical experiments to compare the performance of these methods with an existing explicit stabilized method of weak order 2.

We propose new explicit exponential Runge--Kutta methods for the weak approximation of solutions of stiff Itô stochastic differential equations (SDEs). We also consider the use of exponential Runge--Kutta methods in combination with splitting methods. These methods have weak order 2 for multidimensional, noncommutative SDEs with a semilinear drift term, whereas they are of order 2 or 3 for semilinear ordinary differential equations. These methods are A-stable in the mean square sense for a scalar linear test equation whose drift and diffusion terms have complex coefficients. We carry out numerical experiments to compare the performance of these methods with an existing explicit stabilized method of weak order 2.