William E. Schiesser

The numerical method of lines : integration of partial differential equations

What Is the Numerical Method of Lines? Some Applications of the Numerical Method of Lines. Spatial Differentiation. Initial Value Integration. Stability of Numerical Method of Lines Approximations. Additional Applications: Multidimensional Pdes and Adaptive Grids. Appendix A: The Laplacian Operator in Various Coordinate Systems. Appendix B: Spatial Differentiation Routines. Appendix C: Library of ODE and ODE/PDE Applications. Index.

Adaptive Method of Lines

Introduction Application of the Adaptive Method of Lines to Nonlinear Wave Propagation Problems Adaptive MOL for Magneto-Hydrodynamic PDE Models Development of a 1-D Error-Minimizing Moving Adaptive Grid Method An Adaptive Method of Lines Approach for Modelling Flow and Transport in Rivers An Adaptive Mesh Algorithm for Free Surface Flows in General Geometries, M. Sussman The Solution of Steady PDEs on Adjustable Meshes in Multidimensions Using Local Descent Methods Adaptive Linearly Implicit Methods for Heat and Mass Transfer Problems Linearly Implicit Adaptive Schemes for Singular Reaction-Diffusion Equations Unstructured Mesh MOL Solvers for Reacting Flow Problems Two-Dimensional Model of a Reaction Bonded Aluminum Oxide Cylinder Method of Lines within the Simulation Environment DIVA for Chemical Processes.

Partial differential equation

In mathematics, partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic. They find their generalization in stochastic partial differential equations. Just as ordinary differential equations often model dynamical systems, partial differential equations often model multidimensional systems.

Dynamic Modeling of Transport Process Systems

The Nature of Dynamic Systems. Basic Concepts in the Numerical Integration of Ordinary Differential Equations. Accuracy in the Numerical Integration of Ordinary Differential Equations. Stability in the Numerical Integration of Ordinary Differential Equations. Systems Modeled by Ordinary Differential Equations. Systems Modeled by First Order Partial Differential Equations. Systems Modeled by Second Order Partial Differential Equations. Systems Modeled by First/Second Order, Multidimensional andMultidomain Partial Differential Equations. Appendices 1-9. Index.

Diffusion of a soluble protein, photoactivatable GFP, through a sensory cilium

Transport of proteins to and from cilia is crucial for normal cell function and survival, and interruption of transport has been implicated in degenerative and neoplastic diseases. It has been hypothesized that the ciliary axoneme and structures adjacent to and including the basal bodies of cilia impose selective barriers to the movement of proteins into and out of the cilium. To examine this hypothesis, using confocal and multiphoton microscopy we determined the mobility of the highly soluble photoactivatable green fluorescent protein (PAGFP) in the connecting cilium (CC) of live Xenopus retinal rod photoreceptors, and in the contiguous subcellular compartments bridged by the CC, the inner segment (IS) and the outer segment (OS). The estimated axial diffusion coefficients are DCC = 2.8 ± 0.3, DIS = 5.2 ± 0.6, and DOS = 0.079 ± 0.009 µm2 s−1. The results establish that the CC does not pose a major barrier to protein diffusion within the rod cell. However, the results also reveal that axial diffusion in each of the rod’s compartments is substantially retarded relative to aqueous solution: the axial diffusion of PAGFP was retarded ∼18-, 32- and 1,000-fold in the IS, CC, and OS, respectively, with ∼20-fold of the reduction in the OS attributable to tortuosity imposed by the lamellar disc membranes. Previous investigation of PAGFP diffusion in passed, spherical Chinese hamster ovary cells yielded DCHO = 20 µm2 s−1, and estimating cytoplasmic viscosity as Daq/DCHO = 4.5, the residual 3- to 10-fold reduction in PAGFP diffusion is ascribed to sub-optical resolution structures in the IS, CC, and OS compartments.

Method of lines

Our physical world is most generally described in scientific and engineering terms with respect to threedimensional space and time which we abbreviate as spacetime. PDEs provide a mathematical description of physical spacetime, and they are therefore among the most widely used forms of mathematics. As a consequence, methods for the solution of PDEs, such as the MOL [Sch-91, Sch-09, Gri-11], are of broad interest in science and engineering.

Method of lines solution of the Korteweg-de vries equation

Abstract The Korteweg-de Vries equation (KdVE) is a classical nonlinear partial differential equation (PDE) originally formulated to model shallow water flow. In addition to the applications in hydrodynamics, the KdVE has been studied to elucidate interesting mathematical properties. In particular, the KdVE balances front sharpening and dispersion to produce solitons, i.e., traveling waves that do not change shape or speed. In this paper, we compute a solution of the KdVE by the method of lines (MOL) and compare this numerical solution with the analytical solution of the kdVE. In a second numerical solution, we demonstrate how solitons of the KdVE traveling at different velocities can merge and emerge. The numerical procedure described in the paper demonstrates the ease with which the MOL can be applied to the solution of PDEs using established numerical approximations implemented in library routines.

Upwinding in the method of lines

The method of lines (MOL) is a procedure for the numerical integration of partial differential equations (PDEs). Briefly, the spatial (boundary value) derivatives of the PDEs are approximated algebraically using, for example, finite differences (FDs). If the PDEs have only one initial value variable, typically time, then a system of initial value ordinary differential equations (ODEs) results through the algebraic approximation of the spatial derivatives.

PDE boundary conditions from minimum reduction of the PDE

Abstract Convection-diffusion partial differential equations (PDEs) are second order in the direction of flow, and therefore require two boundary conditions with respect to the spatial independent variable in that direction. The entering or “inflow” boundary condition is usually obvious. The exiting or “outflow” boundary condition is not so obvious. We propose reducing the PDE by one order in the direction of flow to produce the outflow boundary condition. This procedure is illustrated with a dynamic version of the classical Graetz problem in heat transfer. The proposed procedure has the advantages of physically meaningful boundary conditions which produce numerical solutions of good accuracy, and ease of implementation in a method of lines code.

Fluorescence relaxation in 3D from diffraction-limited sources of PAGFP or sinks of EGFP created by multiphoton photoconversion.

The relaxation of fluorescence from diffraction‐limited sources of photoactivatable green fluorescent protein (PAGFP) or sinks of photobleached enhanced GFP (EGFP) created by multiphoton photo‐conversion was measured in solutions of varied viscosity (η), and in live, spherical Chinese hamster ovary (CHO) cells. Fluorescence relaxation was monitored with the probing laser fixed, or rapidly scanning along a line bisected by the photoconversion site. Novel solutions to several problems that hamper the study of PAGFP diffusion after multiphoton photoconversion are presented. A theoretical model of 3D diffusion in a sphere from a source in the shape of the measured multiphoton point‐spread function was applied to the fluorescence data to estimate the apparent diffusion coefficient, Dap. The model incorporates two novel features that make it of broad utility. First, the model includes the no‐flux boundary condition imposed by cell plasma membranes, allowing assessment of potential impact of this boundary on estimates of Dap. Second, the model uses an inhomogeneous source term that, for the first time, allows analysis of diffusion from sources produced by multiphoton photoconversion pulses of varying duration. For diffusion in aqueous solution, indistinguishable linear relationships between Dap and η−1 were obtained for the two proteins: for PAGFP, Daq= 89 ± 2.4 μm2 s−1 (mean ± 95% confidence interval), and for EGFP Daq= 91 ± 1.8 μm2 s−1. In CHO cells, the application of the model yielded Dap= 20 ± 3 μm2 s−1 (PAGFP) and 19 ± 2 μm2 s−1 (EGFP). Furthermore, the model quantitatively predicted the decline in baseline fluorescence that accompanied repeated photobleaching cycles in CHO cells expressing EGFP, supporting the hypothesis of fluorophore depletion as an alternative to the oft invoked ‘bound fraction’ explanation of the deviation of the terminal fluorescence recovery from its pre‐bleach baseline level. Nonetheless for their identical diffusive properties, advantages of PAGFP over EGFP were found, including an intrinsically higher signal/noise ratio with 488‐nm excitation, and the requirement for ∼1/200th the cumulative light energy to produce data of comparable signal/noise.