numerical solution of hyperbolic partial differential equations pdf

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Ngày tải lên: 21/06/2014, 11:20

... Bruti-Liberati Numerical Solution of Stochastic Differential Equations with Jumps in Finance Eckhard Platen Nicola Bruti-Liberati (1975–2007) School of Finance and Economics Department of Mathematical ... about the numerical solution of stochastic differential equations on the webpage of the first author under Numerical Methods”: http://www.business.uts.edu.au/ finance/staff/Eckhard /Numerical Methods.html ... the existence and uniqueness of solutions of SDEs These tools and results provide the basis for the application and numerical solution of stochastic differential equations with jumps 1.1 Stochastic...

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... Splitting-up Method for Partial Differential Equations and Its Apptications to Navier-Stockes Equations, Applied Mathematics Letters Vol 4, No (1992) 25 Vu Tien Dung / VNU Journal of Science, Mathematics ... number of iterations needed for convergence and the total time for the serial computation of Red - Black SOR and Jacobi method are given in the following tables Table Number of Iterations of sequential ... eight computing nodes of 51.2GFlops Each node contains two Intel Xeon dual core 3.2GHz, 2GB Ram The following table shows the dependence of the error of the approximate solutions on the τ number...

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... What are the variables? • What equations are satisfied in the interior of the region of interest? • What equations are satisfied by points on the boundary of the region of interest? (Here Dirichlet ... conceptually, to the solution of large numbers of simultaneous algebraic equations When such equations are nonlinear, they are usually solved by linearization and iteration; so without much loss of generality ... problem as being the solution of special, large linear sets of equations As an example, one which we will refer to in §§19.4–19.6 as our “model problem,” let us consider the solution of equation (19.0.3)...

Ngày tải lên: 28/10/2013, 22:15

... 838 Chapter 19 Partial Differential Equations stable unstable ∆t ∆t ∆x ∆x x or j (a) ( b) Figure 19.1.3 Courant condition for stability of a differencing scheme The solution of a hyperbolic problem ... coefficients of the difference equations are so slowly varying as to be considered constant in space and time In that case, the independent solutions, or eigenmodes, of the difference equations are all of ... rewritten as 840 Chapter 19 Partial Differential Equations Other Varieties of Error ξ = e−ik∆x + i − v∆t ∆x sin k∆x (19.1.25) An arbitrary initial wave packet is a superposition of modes with different...

Ngày tải lên: 07/11/2013, 19:15

... 19 Partial Differential Equations The physical interpretation of the restriction (19.2.6) is that the maximum allowed timestep is, up to a numerical factor, the diffusion time across a cell of ... Ames, W.F 1977, Numerical Methods for Partial Differential Equations, 2nd ed (New York: Academic Press), Chapter Goldberg, A., Schey, H.M., and Schwartz, J.L 1967, American Journal of Physics, vol ... amplitudes, so that the evolution of the larger-scale features of interest takes place superposed with a kind of “frozen in” (though fluctuating) background of small-scale stuff This answer gives...

Ngày tải lên: 15/12/2013, 04:15

... These will occupy us for the remainder of the chapter CITED REFERENCES AND FURTHER READING: Ames, W.F 1977, Numerical Methods for Partial Differential Equations, 2nd ed (New York: Academic Press) ... PDEs When you increase the order of a differencing method to greater than the order of the original PDEs, you introduce spurious solutions to the difference equations This does not create a problem ... αx sin ky ∆)2 856 Chapter 19 Partial Differential Equations (19.3.13) Called the alternating-direction implicit method (ADI), this embodies the powerful concept of operator splitting or time...

Ngày tải lên: 15/12/2013, 04:15

... level of CR, we have reduced the number of equations by a factor of two Since the resulting equations are of the same form as the original equation, we can repeat the process Taking the number of ... North America) ∂u = g(y) ∂x 862 Chapter 19 Partial Differential Equations The finite-difference form of equation (19.4.28) can be written as a set of vector equations uj−1 + T · uj + uj+1 = gj ∆2 ... somewhat more general; its applicability is related to the question of whether the equations are separable (in the sense of “separation of variables”) Both methods require the boundaries to coincide...

Ngày tải lên: 15/12/2013, 04:15

... ease of programming outweighs expense of computer time Occasionally, the sparse matrix methods of §2.7 are useful for solving a set of difference equations directly For production solution of large ... Chapter 19 Partial Differential Equations ADI (Alternating-Direction Implicit) Method The ADI method of §19.3 for diffusion equations can be turned into a relaxation method for elliptic equations ... left-hand sides of equations (19.5.36) and (19.5.37) are tridiagonal (and usually positive definite), so the equations can be solved by the Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC...

Ngày tải lên: 15/12/2013, 04:15

... is the exact solution of the difference equations The difference between this exact solution and the solution for a small number of cycles is the iteration error Now fix the number of cycles to ... set of linear algebraic equations as Lh uh = fh (19.6.2) Let uh denote some approximate solution to equation (19.6.2) We will use the symbol uh to denote the exact solution to the difference equations ... number of cycles (e.g., [2]), you can easily determine it empirically Fix the finest level and study the solution values as you increase the number of cycles per level The asymptotic value of the solution...

Ngày tải lên: 24/12/2013, 12:16

... proof of the Krein–Milman theorem, see Problem (b) In the theory of partial differential equations Let us mention, for example, that the existence of a fundamental solution for a general differential ... analysis (FA) and partial differential equations (PDEs) The first part deals with abstract results in FA and operator theory The second part concerns the study of spaces of functions (of one or more ... {x ∈ E ; p(x) < 1} ∀x ∈ E, Proof of Lemma 1.2 It is obvious that (1) holds Proof of (9) Let r > be such that B(0, r) ⊂ C; we clearly have p(x) ≤ x r ∀x ∈ E Proof of (10) First, suppose that x...

Ngày tải lên: 04/02/2014, 11:10

... linear equations in two indenpendent variables 3.1 Introduction 3.2 Classification 3.3 Canonical form of hyperbolic equations 3.4 Canonical form of parabolic equations 3.5 Canonical form of elliptic ... Such equations are often called semilinear r Scalar equations versus systems of equations A single PDE with just one unknown function is called a scalar equation In contrast, a set of m equations ... different areas of application The fundamental 1.4 Differential equations as mathematical models laws of physics provide a mathematical description of nature’s phenomena on a variety of scales of time...

Ngày tải lên: 16/02/2014, 15:20

... page intentionally left blank Ordinary Differential Equations CHAPTER 0.1 Homogeneous Linear Equations The subject of most of this book is partial differential equations: their physical meaning, ... with real data A new feature of this edition is a CD with auxiliary materials: animations of convergence of Fourier series; animations of solutions of the heat and wave equations as well as ordinary ... follow the derivations of the heat and wave equations The principal objective of the book is solving boundary value problems involving partial differential equations Separation of variables receives...

Ngày tải lên: 17/02/2014, 14:20

... Every classical solution of Problem IV is also a solution of Problem 1.5.1 On the other hand, every solution of Problem 1.5.1 that is smooth enough is also a classical solution of Problem IV Theorem ... in terms of the velocity field v, the pressure (mean normal stress) p, and the concentration c by means of a system of partial differential equations that are a consequence of the balance of mass, ... an analysis that discusses the nature of the solution We shall however restrict ourselves to the question of existence of solutions in this study Motions of incompressible fluids that are reacting...

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