Tham khảo tài liệu 'chl - a finite element scheme for shock capturing_3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Simpo PDF Merge and Split Unregistered Version - http 45 where Uh F Q If we consider the linearized system with the Jacobian matrix A as a constant the nonconservative shallow-water equations may be written as 46 where A and the subscript 0 indicates a constant value. We may select the matrix p such that p 1A p A where A is the matrix of eigenvalues of A and p and P 1 are composed of the eigenvectors. 17 Chapter 2 Numerical Approach Simpo PDF Merge and Split Unregistered Version - http A ÍH Xj 0 0 2 ĩ o co 0 v0-c0 If we define a new set of variables the Riemann Invariants as T - PQ we may write the shallow-water equations as two decoupled equations for which it is apparent that we can propose a test function as lip z a AX IA A-1 dX which can be returned to the original system in terms of the variable Q as Zip Z j a AX p 1 A A-1P 21 ỞX 48 49 The size and direction of the added odd function is then based upon the magnitude and direction of the characteristics. This particular test function is weighted upstream along characteristics. This is a concept like that developed in the finite difference method of Courant Isaacson and Rees 1952 for one-sided differences. These ideas were expanded to more general problems by Moretti 1979 and Gabutti 1983 as split-coefficient matrix methods and by the generalized flux vector splitting proposed by Steger and Warming 1981 . In the finite elements community instead of one-sided differences the test function is weighted upstream. This particular method in 1-D is equivalent to the SUPG scheme of Hughes and Brooks 1982 and similar to the form proposed by Dendy 1974 . Examples of this approach in the open-channel environment are for the generalized shallow-water equations in 1-D in Berger and Winant 1991 and for 2-D in Berger 1992 . A 1-D St. Venant application is given by Hicks and Steffler 1992 . If we analyze this approach on a uniform grid we find the following roots 18 Chapter 2 Numerical Approach
