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Godunov Scheme
In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by S. K. Godunov in 1959, for solving partial differential equations. One can think of this method as a conservative finite-volume method which solves exact, or approximate Riemann problems at each inter-cell boundary. In its basic form, Godunov's method is first order accurate in both space and time, yet can be used as a base scheme for developing higher-order methods. Basic scheme Following the classical finite-volume method framework, we seek to track a finite set of discrete unknowns, : Q^_i = \frac \int_ ^ q(t^n, x)\, dx where the x_ = x_ + \left( i - 1/2 \right) \Delta x and t^n = n \Delta t form a discrete set of points for the hyperbolic problem: : q_t + ( f( q ) )_x = 0, where the indices t and x indicate the derivations in time and space, respectively. If we integrate the hyperbolic problem over a control volume _, x_ we obtain a me ... [...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 at finding 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, 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 simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Volume Method
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages. "Finite volume" refers to the small volume surrounding each node point on a mesh. Finite volume methods can be compared and contrasted with the finite difference methods, which approximate derivatives using nodal values, or finite element methods, which create local approximations of a soluti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Fluid Dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate the free-stream flow of the fluid, and the interaction of the fluid ( liquids and gases) with surfaces defined by boundary conditions. With high-speed supercomputers, better solutions can be achieved, and are often required to solve the largest and most complex problems. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial validation of such software is typically performed using experimental apparatus such as wind tunnels. In addition, previously performed analytical or empirical analysis of a particular problem can be used for comparison. A final validation is often performed using full-scale testing, such as flight tests. CFD is applied to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advection Upstream Splitting Method
The advection upstream splitting method (AUSM) is developed as a numerical inviscid flux function for solving a general system of conservation equations. It is based on the upwind concept and was motivated to provide an alternative approach to other upwind methods, such as the Godunov method, flux difference splitting methods by Roe, and Solomon and Osher, flux vector splitting methods by Van Leer, and Steger and Warming. The AUSM first recognizes that the inviscid flux consist of two physically distinct parts, i.e., convective and pressure fluxes. The former is associated with the flow (advection) speed, while the latter with the acoustic speed; or respectively classified as the linear and nonlinear fields. Currently, the convective and pressure fluxes are formulated using the eigenvalues of the flux Jacobian matrices. The method was originally proposed by Liou and Steffen for the typical compressible aerodynamic flows, and later substantially improved in Liou, M.-S., “A Seque ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lax–Wendroff Theorem
In computational mathematics, the Lax–Wendroff theorem, named after Peter Lax and Burton Wendroff, states that if a conservative numerical scheme for a hyperbolic system of conservation laws converges, then it converges towards a weak solution. See also * Lax–Wendroff method * Godunov's scheme References * Randall J. LeVeque Randall J. LeVeque is a Professor of Applied Mathematics at University of Washington who works in many fields including numerical analysis, computational fluid dynamics, and mathematical theory of conservation laws. Among other contributions, he i ..., Numerical methods for conservation laws, Birkhäuser, 1992 Numerical differential equations Computational fluid dynamics Theorems in analysis {{Mathapplied-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Variation Diminishing
In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of this method is in computational fluid dynamics. The concept of TVD was introduced by Ami Harten. Model equation In systems described by partial differential equations, such as the following hyperbolic advection equation, :\frac + a\frac = 0, the total variation (TV) is given by :TV(u(\cdot,t)) = \int \left, \frac \ \mathrmx , and the total variation for the discrete case is, :TV(u^n) = TV(u(\cdot,t^n)) = \sum_j \left, u_^n - u_j^n \ . where u_^n=u(x_,t^n). A numerical method is said to be total variation diminishing (TVD) if, :TV \left( u^\right) \leq TV \left( u^\right) . Characteristics A numerical scheme is said to be monotonicity preserving if the following properties are maintained: *If u^ is monotonically increasing (or decreasing) in space, then so is u^. proved the follow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lax–Friedrichs Method
The Lax–Friedrichs method, named after Peter Lax and Kurt O. Friedrichs, is a numerical method for the solution of hyperbolic partial differential equations based on finite differences. The method can be described as the FTCS (forward in time, centered in space) scheme with a numerical dissipation term of 1/2. One can view the Lax–Friedrichs method as an alternative to Godunov's scheme, where one avoids solving a Riemann problem at each cell interface, at the expense of adding artificial viscosity. Illustration for a Linear Problem Consider a one-dimensional, linear hyperbolic partial differential equation for u(x,t) of the form: : u_t + au_x = 0\, on the domain : b \leq x \leq c,\; 0 \leq t \leq d with initial condition : u(x,0) = u_0(x)\, and the boundary conditions : u(b,t) = u_b(t)\, : u(c,t) = u_c(t).\, If one discretizes the domain (b, c) \times (0, d) to a grid with equally spaced points with a spacing of \Delta x in the x-direction and \Delta t in the t-dir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
High-resolution Scheme
High-resolution schemes are used in the numerical solution of partial differential equations where high accuracy is required in the presence of shocks or discontinuities. They have the following properties: *Second- or higher-Order of accuracy, order spatial accuracy is obtained in smooth parts of the solution. *Solutions are free from spurious oscillations or wiggles. *High accuracy is obtained around shocks and discontinuities. *The number of mesh points containing the wave is small compared with a first-order scheme with similar accuracy. General methods are often not adequate for accurate resolution of steep gradient phenomena; they usually introduce non-physical effects such as ''smearing'' of the solution or ''spurious oscillations''. Since publication of ''Godunov's order barrier theorem'', which proved that linear methods cannot provide non-oscillatory solutions higher than first order (Godunov 1954, Godunov 1959), these difficulties have attracted much attention and a nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Godunov's Theorem
In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high resolution schemes for the numerical solution of partial differential equations. The theorem states that: :''Linear numerical schemes for solving partial differential equations (PDE's), having the property of not generating new extrema (monotone scheme), can be at most first-order accurate.'' Professor Sergei K. Godunov originally proved the theorem as a Ph.D. student at Moscow State University. It is his most influential work in the area of applied and numerical mathematics and has had a major impact on science and engineering, particularly in the development of methods used in computational fluid dynamics (CFD) and other computational fields. One of his major contributions was to prove the theorem (Godunov, 1954; Godunov, 1959), that bears his name. The theorem We generally fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MUSCL Scheme
In the study of partial differential equations, the MUSCL scheme is a finite volume method that can provide highly accurate numerical solutions for a given system, even in cases where the solutions exhibit shocks, discontinuities, or large gradients. MUSCL stands for ''Monotonic Upstream-centered Scheme for Conservation Laws'' (van Leer, 1979), and the term was introduced in a seminal paper by Bram van Leer (van Leer, 1979). In this paper he constructed the first ''high-order'', ''total variation diminishing'' (TVD) scheme where he obtained second order spatial accuracy. The idea is to replace the piecewise constant approximation of Godunov's scheme by reconstructed states, derived from cell-averaged states obtained from the previous time-step. For each cell, slope limited, reconstructed left and right states are obtained and used to calculate fluxes at the cell boundaries (edges). These fluxes can, in turn, be used as input to a ''Riemann solver'', following which the solutions are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Matrix And Determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. Suppose is a function such that each of its first-order partial derivatives exist on . This function takes a point as input and produces the vector as output. Then the Jacobian matrix of is defined to be an matrix, denoted by , whose th entry is \mathbf J_ = \frac, or explicitly :\mathbf J = \begin \dfrac & \cdots & \dfrac \end = \begin \nabla^ f_1 \\ \vdots \\ \nabla^ f_m \end = \begin \dfrac & \cdots & \dfrac\\ \vdots & \ddots & \vdots\\ \dfrac & \cdots ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Courant–Friedrichs–Lewy Condition
In mathematics, the convergence condition by Courant–Friedrichs–Lewy is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. It arises in the numerical analysis of explicit time integration schemes, when these are used for the numerical solution. As a consequence, the time step must be less than a certain time in many explicit time-marching computer simulations, otherwise the simulation produces incorrect results. The condition is named after Richard Courant, Kurt Friedrichs, and Hans Lewy who described it in their 1928 paper. Heuristic description The principle behind the condition is that, for example, if a wave is moving across a discrete spatial grid and we want to compute its amplitude at discrete time steps of equal duration, then this duration must be less than the time for the wave to travel to adjacent grid points. As a corollary, when the grid point separation is reduced, the upper limit for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
