High order operator splitting methods based on an integral deferred correction framework

Integral deferred correction (IDC) methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties. In this paper, we construct high order operator splitting schemes using the IDC procedure to solve initial value problems (IVPs). We present analysis to show that the IDC methods can correct for both the splitting and numerical errors, lifting the order of accuracy by $r$ with each correction, where $r$ is the order of accuracy of the method used to solve the correction equation. We further apply this framework to solve partial differential equations (PDEs). Numerical examples in two dimensions of linear and nonlinear initial-boundary value problems are presented to demonstrate the performance of the proposed IDC approach.Comment: 33 pages, 22 figure

Operator Splitting Monte Carlo Method for Aerosol Dynamics

Aerosol dynamics are described by the population balance equation (PBE). In principle, three typical methods (i.e., direct discretization, method of moments, and stochastic method) have been widely used to solve the PBE. Stochastic method is the most flexible among the three methods. However, stochastic method is computationally expensive. Recently, an operator splitting Monte Carlo (OSMC) method has been developed so as to improve the numerical efficiency while preserving the flexibility of the stochastic method. Within the OSMC, nucleation and surface growth are handled with deterministic means, while coagulation is simulated with a stochastic method (the Marcus‐Lushnikov stochastic process). The stochastic and deterministic treatments of various aerosol dynamic processes are synthesized under the framework of operator splitting. Here, the operator splitting errors of various schemes are analyzed rigorously, combined with concrete numerical examples. The analyses not only provide sound theoretical bases for selecting the most efficient operator splitting scheme for the usage of the OSMC, but also shed lights on how to adopt operator splitting in other PBE solving methods, i.e., direct discretization, method of moments, etc

OPERATOR SPLITTING METHOD AND PROPAGATION FAILURE IN REACTION-DIFFUSION EQUATIONS

The focus of this dissertation is to study the solution behavior of the Nagumo type reaction-diffusion equation that is known for modeling impulse propagation in nerve axons. The study involves working with an operator splitting scheme along with an innovative regularization idea. The use of an operator splitting scheme helps in de-constructing a problem into simpler subproblems. Our work involves studying the the forward and inverse problems of the Nagumo equation. We also extend our study to understand pinning or propagation failure in the discrete version of the Nagumo equation. We implement the numerical scheme while maintaining the stability and accuracy of the full problem

Path Reweighting Methods for underdamped Langevin Dynamics for Molecular Systems

Knowledge about the dynamical properties of biomolecules is essential to understand their function in biological processes. This thesis approaches the task to compute dynamical properties with two different strategies. Part A focuses on Molecular Dynamics (MD) simulations combined with path reweighting. Three of the most widely used underdamped Langevin integrators for MD simulations are the splitting methods BAOAB and BAOA which are available in the MD packages OpenMM and AMBER and the Gromacs Stochastic Dynamics (GSD) integrator implemented in GROMACS. We found that all three integrators are equivalent configurational sampling algorithms and thus yield configurational properties at equivalent accuracy. MD simulations with stochastic integrators such as Langevin integrators offer the possibility to reweight estimated dynamical properties using path reweighting. With path reweighting we can for example recover the original dynamics from MD simulation that have been conducted with enhanced sampling methods. The key component of path reweighting is the path reweighting factor M which strongly depends on the chosen integrator. We derive M_L for underdamped Langevin dynamics propagated by a variant of the Langevin Leapfrog integrator. Additionally, we present two strategies which can be used as blueprints to straightforwardly derive M_L for other Langevin integrators. The previously reported path reweighting factor matches the Euler-Maruyama integrator for overdamped Langevin dynamics and was used as standard reweighting factor even though the MD simulation was conducted with an underdamped Langevin integrator. We prove that this path reweighting factors differs from the exact M_L only by O(ξ^4 ∆t^4) and thus yields highly accurate dynamical reweighting results (∆t is the integration time step, and ξ is the collision rate.). Part B of this thesis combines experimental and theoretical approaches to investigate Multiple Inositol Polyphosphate Phosphatase 1 (MINPP1)-mediated inositol polyphosphate (InsP) networks. We use 13C-labeling experiments combined with nuclear magnetic resonance spectroscopy (NMR) to uncover a novel branch of InsP dephosphorylation in human cells. Additionally, we extract the corresponding reaction rates using a Markovian kinetic scheme as theoretical model to describe the network

Modelado y simulación de procesos sedimentarios en aguas someras

En esta tesis se deduce un modelo matemático de tipo aguas someras para el transporte de sedimentos sin hacer uso de la hipótesis de Boussinesq. Dicho modelo es resuelto de forma numerica mediante un esquema robusto y eficiente, el cuál, permite garantizar la positividad de todas las variables conservadas. Las técnicas descritas son validadas mediante comparación con datos experimentales. Muchos de los modelos para el transporte de sedimentos, al ser deducidos, utilizan la hipótesis de Boussinesq, esto implica que se puede considerar que la distribución de la mezcla del agua y del sedimento es casi constante, lo que ayuda a que el modelado sea más sencillo. No obstante, esta simplificación puede ocasionar imprecisiones en los resultados obtenidos para muchas situaciones reales La solución numérica del sistema deducido se hace mediante un algoritmo de dos pasos. En el primero, se resuelve la parte hiperbólica del problema siguiendo la idea de los esquemas numéricos de volúmenes-finitos camino-conservativo. En particular, se ha desarrollado un esquema tipo matriz polinomial de viscosidad (PVM) de tres ondas. En el segundo paso, se resuelven los términos fuentes de erosión, depósito y fricción de manera semi-implícita. La forma que se propone para la resolución de dichos términos fuentes, permite probar de forma sencilla la positividad del fondo sedimentado y de la densidad de la mezcla. Con lo cual tenemos un esquema numérico que preserva la positividad globalmente. El modelo matemático y el esquema numérico son validados mediante experimentos numéricos, unidimensionales y bidimensionales, comparando los resultados con datos obtenidos en laboratorio