On an effective solution of the Riemann problem for the second-order improperly elliptic equation in the rectangle
© Babayan and Raeisian; licensee Springer 2013
Received: 12 September 2012
Accepted: 31 May 2013
Published: 28 June 2013
In this paper we present the numerical method for the solution of the Riemann problem for the second-order improperly elliptic equation. First, we reduce this problem to boundary value problems for properly elliptic equations, and after that we solve these problems by the grid method.
MSC:35G45, 35G15, 35J25, 35J57, 65N06, 65N20.
reduces to ill-conditioned linear system with zeroes on the main diagonal, so we introduce another variant of the solution.
2 Description of the algorithm of solution
Here a function is the already known function . After determining the function ψ, we get the boundary value problem, analogous to (13), to determine the imaginary part of the function Ψ.
Thus, the main idea of this algorithm is to reduce the previous problem for an improperly elliptic equation to the boundary value problems for properly elliptic equations (in our case, to boundary value problems for biharmonic and Laplace equations). In the next section we describe the realization of this algorithm by the grid method.
3 Solution of problem (3), (4) by the grid method
and is a standard grid analogue (forward divided difference) of the operator , and are values of the functions f and g in boundary points of the grid. This problem approximates problem (9), and the rate of approximation is for a biharmonic equation and for boundary conditions (see ). Therefore, from the stability of problem (18), we get the convergence of the grid function to (, p.30, Theorem 2.5). From the last two equations of (18), we get the values of the function in the points for ; and ; , and we find the values in interior nodes from the linear system with a symmetric pentadiagonal matrix. Hence, we get the stability of (18) from the positive definiteness of the main matrix of this system, and an algorithm for the solution of this system may be found in .
in the grid points, here , , are the real constants. And at the final step, we must find the function Ψ. First, we solve the Dirichlet problem for the Laplace equation to determine the real part of Ψ, and then we determine the imaginary part of Ψ solving the problem analogous to (13). Here we get the last arbitrary constant .
Summing up, we can say that the previous problem may be reduced to boundary value problems for properly elliptic equations.
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