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Concept# Numerical methods for partial differential equations

Summary

Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).
In principle, specialized methods for hyperbolic, parabolic or elliptic partial differential equations exist.
Finite difference method
In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.
Method of lines
The method of lines (MOL, NMOL, NUMOL) is a technique for solving partial differential equations (PDEs) in which all dimensions except one are discretized. MOL allows standard, general-purpose methods and software, developed for the numerical integration of ordinary differential equations (ODEs) and differential algebraic equations (DAEs), to be used. A large number of integration routines have been developed over the years in many different programming languages, and some have been published as open source resources.
The method of lines most often refers to the construction or analysis of numerical methods for partial differential equations that proceeds by first discretizing the spatial derivatives only and leaving the time variable continuous. This leads to a system of ordinary differential equations to which a numerical method for initial value ordinary equations can be applied. The method of lines in this context dates back to at least the early 1960s.
Finite element method
The finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for differential equations. It uses variational methods (the calculus of variations) to minimize an error function and produce a stable solution. Analogous to the idea that connecting many tiny straight lines can approximate a larger circle, FEM encompasses all the methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain.

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Numerical methods for partial differential equations

Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). In principle, specialized methods for hyperbolic, parabolic or elliptic partial differential equations exist. Finite difference method In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.

Finite difference method

In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.

Additive Schwarz method

In mathematics, the additive Schwarz method, named after Hermann Schwarz, solves a boundary value problem for a partial differential equation approximately by splitting it into boundary value problems on smaller domains and adding the results. Partial differential equations (PDEs) are used in all sciences to model phenomena. For the purpose of exposition, we give an example physical problem and the accompanying boundary value problem (BVP). Even if the reader is unfamiliar with the notation, the purpose is merely to show what a BVP looks like when written down.

Related lectures (713)

Claudia Maria Colciago, Anna Scotti

In the last years, numerous seismological evidences have shown a strict correlation between fluid injection and seismicity. An important topic that is currently under discussion in the scientific community concerns the prediction of the earthquake magnitude that may be triggered by fluid injection activities. Coupled fluid flow and geomechanical deformation models can aim at understanding the evolution of pore pressure and rock deformation due to fluid injection in the subsurface. To perform an accurate numerical study of the correlation among fluid injection, seismicity rates and maximum earthquake magnitude, it is necessary to characterize the model with two fundamental features: first, the presence of a system of faults possibly intersecting among each other; second, the variability of the hydro-mechanical properties across the region surrounding each fault plane (fault zone). The novelty of this work is to account for these two aspects combining together two different numerical techniques that have been proposed in literature for the fault's modelling: for the first feature, interface elements are used to describe the frictional contacts occurring on the fault surfaces; for the second feature, solid elements are adopted to describe the heterogeneous hydro-mechanical behavior across the fault zone. Moreover, we account for a spatial variation of the permeability in the fault zone both along the dip and the normal direction with respect to the fault plane. We compute the numerical solution for six models among which we vary the permeability contrasts between protolith rocks and damage zone and between damage zone and fault core. We demonstrate that the anisotropy of permeability of the fault zone has a strong impact both on the timing and on the magnitude of triggered earthquakes. We suggest that a similar approach, which includes the entire architecture of the fault zone, shall be included in fluid-flow-geomechanical simulations to improve fault stability analysis during fluid injection.

The modeling of a system composed by a gas phase and organic aerosol particles, and its numerical resolution are studied. The gas-aerosol system is modeled by ordinary differential equations coupled with a mixed-constrained optimization problem. This coupling induces discontinuities when inequality constraints are activated or deactivated. Two approaches for the solution of the optimization-constrained differential equations are presented. The first approach is a time splitting scheme together with a fixed-point method that alternates between the differential and optimization parts. The ordinary differential equations are approximated by the Crank-Nicolson scheme and a primal-dual interior-point method combined with a warm-start strategy is used to solve the minimization problem. The second approach considers the set of equations as a system of differential algebraic equations after replacing the minimization problem by its first order optimality conditions. An implicit 5th-order Runge-Kutta method (RADAU5) is then used. Both approaches are completed by numerical techniques for the detection and computation of the events (activation and deactivation of inequality constraints) when the system evolves in time. The computation of the events is based on continuation techniques and geometric arguments. Moreover the first approach completes the computation with extrapolation polynomials and sensitivity analysis, whereas the second approach uses dense output formulas. Numerical results for gas-aerosol system made of several chemical species are proposed for both approaches. These examples show the efficiency and accuracy of each method. They also indicate that the second approach is more efficient than the first one. Furthermore theoretical examples show that the method for the computation of the activation is of second order for the first approach and exact for the second one.

In this work a physical modelling framework is presented, describing the intelligent, non-local, and anisotropic behaviour of pedestrians. Its phenomenological basics and constitutive elements are detailed, and a qualitative analysis is provided. Within this common framework, two first-order mathematical models, along with related numerical solution techniques, are derived. The models are oriented to specific real world applications: a one-dimensional model of crowd-structure interaction in footbridges and a two-dimensional model of pedestrian flow in an underground station with several obstacles and exits. The noticeable heterogeneity of the applications demonstrates the significance of the physical framework and its versatility in addressing different engineering problems. The results of the simulations point out the key role played by the physiological and psychological features of human perception on the overall crowd dynamics. (C) 2010 Elsevier Inc. All rights reserved.

2011Finite Element Method: Basics and ApplicationsME-484: Numerical methods in biomechanics

Introduces the Finite Element Method for solving PDEs and demonstrates its application through examples and Comsol Multiphysics.

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Explores the Radiative Transport Equation in tissue optics, covering radiance, photon distribution, fluence rate, and solution methods.