NAME, COMPUTER, PROBLEM, SOLUTION, RESTRICTIONS, CPU, FEATURES, STATUS, REFERENCES, REQUIREMENTS, LANGUAGE, OPERATING SYSTEM, OTHER RESTRICTIONS, AUTHOR, MATERIAL, CATEGORIES

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Program name | Package id | Status | Status date |
---|---|---|---|

2DEPEP | NESC0806/01 | Tested | 23-DEC-1980 |

Machines used:

Package ID | Orig. computer | Test computer |
---|---|---|

NESC0806/01 | CDC 7600 | CDC 7600 |

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3. DESCRIPTION OF PROGRAM OR FUNCTION

2DEPEP solves the partial differential equation system:

C1(X,Y,U,V,T)*DU/DT=D(OXX)/DX+D(OXY)/DY+F1(X,Y,U,V,T)

C2(X,Y,U,V,T)*DV/DT=D(OYX)/DX+D(OYY)/DY+F2(X,Y,U,V,T) in a general two-dimensional region, R, with

U=FB1(S)

V=FB2(S) for S on BR1, and

OXX*NX+OXY*NY=GB1(S,U,V,T)

OYX*NX+OYY*NY=GB2(S,U,V,T) for S on BR2, where BR1 and BR2 are distinct parts of the boundary. (NX,NY)= unit outward normal.

U=U0(X,Y)

V=V0(X,Y) for T=T0, and

OXX=OXX(X,Y,DU/DX,DU/DY,DV/DX,DV/DY,T)

OXY=OXY(X,Y,DU/DX,DU/DY,DV/DX,DV/DY,T)

OYX=OYX(X,Y,DU/DX,DU/DY,DV/DX,DV/DY,T)

OYY=OYY(X,Y,DU/DX,DU/DY,DV/DX,DV/DY,T).

The Jacobian matrices of derivatives of OXX,OXY,OYX,OYY with respect to DU/DX,DU/DY,DV/DX,DV/DY and of F1,F2 with respect to U,V and of GB1,GB2 with respect to U,V must be symmetric.

The related elliptic and eigenvalue problems are also solved by 2DEPEP and single equations can be handled efficiently. Examples of applications of the program are elasticity, one- or two- component diffusion, heat conduction, minimal surface, potential problems and the time-independent and time-dependent Schrodinger equations.

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4. METHODS

The user supplies an initial triangulation which defines the region R. This triangulation is refined as controlled by a user-supplied function. For singular elliptic problems, optimal order convergence is possible if this function approximates the function D3MAX(UV), which is the maximum of the third derivatives of U and V.

The problem is discretized by Galerkin's method, using a trial function space of piecewise quadratic polynomials with respect to the triangulation. An approximate solution is calculated, using either the implicit or the Crank-Nicolson method for each time- step. The non-linear equations which must be solved each time- step are solved by Newton's method. One iteration is sufficient since the solution on the previous time-step is used for starting values. In each application of Newton's method the symmetric, banded Jacobian matrix is inverted directly by Gaussian elimination, with the matrix stored out-of-core when necessary, according to the frontal method. For linear time-independent problems the Jacobian is the same each time-step, so the Cholesky factorization done in the first step is used on all following steps. The isoparametric method is used to handle curved boundaries.

When C1=C2=0 and all functions are independent of T, that is, when the problem is a steady-state (elliptic) problem, each time- step corresponds to one iteration of Newton's method. (One iteration is sufficient for linear problems.) An eigenvalue problem can be converted to a related parabolic problem where each time-step is equivalent to an iteration of the inverse power method for finding the smallest eigenvalue and the corresponding eigenfunction. A single equation can be handled with no loss of efficiency in storage or execution time.

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5. RESTRICTIONS ON THE COMPLEXITY OF THE PROBLEM

At most two simultaneous partial differential equations can be solved. The input data set is limited to 200 cards. This can be increased by changing the value of the variable MXCARD and increasing the dimensions of the arrays L, INDX, and LNAM in the preprocessor.

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NESC0806/01, included references:

- Granville Sewell:2DEPEP, 2-D Elliptic, Parabolic and Eigenvalue Problems, User's

Manual

Purdue University (October 1978).

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NESC0806/01

File name | File description | Records |
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NESC0806_01.001 | INFORMATION FILE | 4 |

NESC0806_01.002 | PREPROCESSOR SOURCE | 538 |

NESC0806_01.003 | TDEPEP SOURCE | 1061 |

NESC0806_01.004 | PROBLEM 1 OUTPUT | 56 |

NESC0806_01.005 | PROBLEM 2 OUTPUT | 174 |

NESC0806_01.006 | PROBLEM 3 OUTPUT | 120 |

NESC0806_01.007 | PROBLEM 4 OUTPUT | 160 |

NESC0806_01.008 | PROBLEM 1 INPUT | 59 |

NESC0806_01.009 | PROBLEM 2 INPUT | 31 |

NESC0806_01.010 | PROBLEM 3 INPUT | 43 |

NESC0806_01.011 | PROBLEM 4 INPUT | 38 |

NESC0806_01.012 | JCL | 70 |

Keywords: banded matrix, eigenvalues, finite element method, iterative methods, partial differential equations, two-dimensional.