Convergence study on numerical methods solving Hyperbolic PDEs (双曲型pde 的数值方法稳定性研究）

          重读大部头书《 Finite Volume Methods for Hyperbolic Problems》。现在知识碎片太多，重读大书能助提纲携领。

         * Hyperbolicalicity of linear systems

         q_t + A q_x =0 

        matrix A is diagonalizable with real eigenvalues, do diagonalization --> w_t + D w_x = 0, where D is diagonal matrix, now the system is decoupled. --> 

        w_t ^ p + \lamba w_x^p = 0

        -->  solutions of original system now can be consisted of a linear combination of m "waves" traveling at characteristic speeds ( \lamba ).

      Define characteristic curves  X(t) = x_0 + \lamba^p t ,  along which the information propagates

     

      *Domain of dependence & Range of influence 

       - Let <X,T> be fixed point in space-time,  then the solution w(X,T) depends only on some particular points ( X - \lamba^p T) from the initial line. (p = 1, ... m), this is called the domain of dependence of <X,T>. on the contrary way, think about which particles will be influenced by the current <X,T> will give the range of influence as < X + \lamba^p T> .

       - For hyperbolic PDE, domain of dependence is always bounded, resulting form the fact that information propagates at finite speed. In design of numerical method means that the explicit methods will be efficiently. While for parabolic PDE, domain of dependence is the entire initial line, so implicit methods is needed.

       - For discontinuous in initial data, only propagate along characteristics for a linear system. meaning, the discontinuous in initial values will extension only along characteristic curves.

        * General Consideration about Convergence and Stability

       - consistent, numerical solution should approximate well locally   | y_{i+1} - y_i | <= L (x_{i+1} - x_i )

      - stable in some sense (L^2 norm usually), the small errors made in each time step do not grow too fast in later time steps.

      -  consistence + stability == convergence 

     

      * CFL conditional number

      Define:  a numerical method can be convergent only if its numerical domain of dependence contains the true domain of dependence of the original PDE.

     Define: domain of dependence of a numerical approximation

     -  let dt/dx = r fixed, for point <X, T> in space-time , the domain of dependence lie in the initial line as 

    0 - T = r ( x- X) 

    0 - T = -r (x - X)

    namely,  [X - T/r,  X + T/r] 

    the domain of depend of a original PDE  is as above ,  D(X, T) = { X - \lamba^p T ; p = 1, 2, .... m } 

    

    so CFL requires         X - T/r <=  X - \lamba^p T <=  X  - T/r

   define Courant Number  \mu =  dt/dx max| \lamba^p| <= 1 

  

   CFL is necessary for convergence, another stability analysis methods is by Von Neuman, by define amplification factor, represent each frequency in the solution is amplified in advancing the solution one time step.

  Von Neuman 方法，在稀疏矩阵迭代方法的收敛性分析中也有应用，既迭代矩阵的谱半径不超过1。 下次再复习下parabolic pde 的数值解性质。 这样把基础理论过一遍。