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FTCS Scheme
In numerical analysis, the FTCS (forward time-centered space) method is a finite difference method used for numerically solving the heat equation and similar parabolic partial differential equations. It is a first-order method in time, explicit in time, and is conditionally stable when applied to the heat equation. When used as a method for advection equations, or more generally hyperbolic partial differential equations, it is unstable unless artificial viscosity is included. The abbreviation FTCS was first used by Patrick Roache. The method The FTCS method is based on the forward Euler method in time (hence "forward time") and central difference in space (hence "centered space"), giving first-order convergence in time and second-order convergence in space. For example, in one dimension, if the partial differential equation is :\frac = F\left(u, x, t, \frac\right) then, letting u(i \,\Delta x, n\, \Delta t) = u_^\,, the forward Euler method is given by: :\frac = F_^\left(u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, as in spinodal decomposition. Diffusion is a stochastic process due to the inherent randomness of the diffusing entity and can be used to model many real-life stochastic scenarios. Therefore, diffusion and the corresponding mathematical models are used in several fields beyond physics, such as statistics, probability theory, information theory, neural networks, finance, and marketing. The concept of diffusion is widely used in many fields, including physics (Molecular diffusion, particle diffusion), chemistry, biology, sociology, economics, statistics, data science, and finance (diffusion of people, ideas, data and price v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite-difference Time-domain Method
Finite-difference time-domain (FDTD) or Yee's method (named after the Chinese American applied mathematician Kane S. Yee, born 1934) is a numerical analysis technique used for modeling computational electrodynamics. History Finite difference schemes for time-dependent partial differential equations (PDEs) have been employed for many years in computational fluid dynamics problems, including the idea of using centered finite difference operators on staggered grids in space and time to achieve second-order accuracy. The novelty of Yee's FDTD scheme, presented in his seminal 1966 paper, was to apply centered finite difference operators on staggered grids in space and time for each electric and magnetic vector field component in Maxwell's curl equations. The descriptor "Finite-difference time-domain" and its corresponding "FDTD" acronym were originated by Allen Taflove in 1980. Since about 1990, FDTD techniques have emerged as primary means to computationally model many scientific and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crank–Nicolson Method
In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a Big O notation, second-order method in time. It is Explicit and implicit methods, implicit in time, can be written as an implicit Runge–Kutta method, and it is Numerical stability, numerically stable. The method was developed by John Crank and Phyllis Nicolson in the 1940s. For diffusion equations (and many other equations), it can be shown the Crank–Nicolson method is unconditionally Numerical stability, stable. However, the approximate solutions can still contain (decaying) spurious oscillations if the ratio of time step \Delta t times the thermal diffusivity to the square of space step, \Delta x^2, is large (typically, larger than 1/2 per Von Neumann stability analysis). For this reason, whenever large time steps or high spatial resolution is necessary, the less accurate backward Euler method is o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabolic Differential Equation
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena in, for example, engineering science, quantum mechanics and financial mathematics. Examples include the heat equation, time-dependent Schrödinger equation and the Black–Scholes equation. Definition To define the simplest kind of parabolic PDE, consider a real-valued function u(x, y) of two independent real variables, x and y. A second-order, linear, constant-coefficient PDE for u takes the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + F = 0, where the subscripts denote the first- and second-order partial derivatives with respect to x and y. The PDE is classified as ''parabolic'' if the coefficients of the principal part (i.e. the terms containing the second derivatives of u) satisfy the condition :B^2 - AC = 0. Usually x represents one-dimensional position and y represents time, and the PDE is solved subje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffusion Equation
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion). In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of the convection–diffusion equation when bulk velocity is zero. It is equivalent to the heat equation under some circumstances. Statement The equation is usually written as: \frac = \nabla \cdot \big D(\phi,\mathbf) \ \nabla\phi(\mathbf,t) \big where is the density of the diffusing material at location and time and is the collective diffusion coefficient for density at location ; and represents the vector differential operator del. If the diffusion coefficient depends on the density then the equatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advection
In the fields of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is also a fluid. The properties that are carried with the advected substance are conserved properties such as energy. An example of advection is the transport of pollutants or silt in a river by bulk water flow downstream. Another commonly advected quantity is energy or enthalpy. Here the fluid may be any material that contains thermal energy, such as water or air. In general, any substance or conserved extensive quantity can be advected by a fluid that can hold or contain the quantity or substance. During advection, a fluid transports some conserved quantity or material via bulk motion. The fluid's motion is described mathematically as a vector field, and the transported material is described by a scalar field showing its distribution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Differential Equation
In mathematics, a linear differential equation is a differential equation that is linear equation, linear in the unknown function and its derivatives, so it can be written in the form a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x) where and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable . Such an equation is an ordinary differential equation (ODE). A ''linear differential equation'' may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. Types of solution A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of antiderivative, integrals. This is also true for a linear equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Partial Differential Equations
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n - 1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is \frac = c^2 \frac The equation has the property that, if and its first time derivative are arbitrarily specified initial data on the line (with sufficient smoothness properties), then there exists a solution for all time . The solutions of hyperbolic equations are "wave-like". If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. Relative to a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context: one important context is numerical linear algebra, and another is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Stability Analysis
In numerical analysis, von Neumann stability analysis (also known as Fourier stability analysis) is a procedure used to check the stability of finite difference schemes as applied to linear partial differential equations. The analysis is based on the Fourier decomposition of numerical error and was developed at Los Alamos National Laboratory after having been briefly described in a 1947 article by British researchers John Crank and Phyllis Nicolson. This method is an example of explicit time integration where the function that defines governing equation is evaluated at the current time. Later, the method was given a more rigorous treatment in an article co-authored by John von Neumann. Numerical stability The stability of numerical schemes is closely associated with numerical error. A finite difference scheme is stable if the errors made at one time step of the calculation do not cause the errors to be magnified as the computations are continued. A ''neutrally stable scheme'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
