Group 4 (PT2)

Saifulhadi-Azri-Faisal-Arif

Objective

• Able to identify the difference between matrix solver and iterative method

• Able to compare them in terms of application in reservoir simulation

Comparison Method Iterative Method Direct Matrix Solver

Definition The mathematical procedure that generates a sequence of improving approximate solutions for a class of problems with additional stopping criteria.

The direct solution methods are characterized by their capacity to produce the solution vector for a given system of linear equations after a fixed number of operations.

Example •Gauss-Seidel •Newton-Raphson •Convergent Matrix •Conjugate gradient •Jacobian Method •Matrix Splitting

•Gaussian Elimination •Domain decomposition •Thomas algorithm •Tang’s algorithm

Application •Decreasing space computation in database •Used to converge to root solution.

•1D Rectangular or Radial Flow Problems (Thomas' Algorithm) •2D and 3D Flow Problems (Sparse Matrices) •Solving last part of Reservoir Numerical simulator.

Comparison

Method Iterative Matrix Solver

Advantages •decrease the data size of database used in simulation. •able to converge to root solution.

•can achieve to the solution directly.

Disadvantages •might not work (converge) then the initial guess is an outlier or having a big timestep. •need a lot of run depending on method (loop)

•different matrix have different solution scheme. •use bigger space in database (according to martix size)

References

• http://faculty.washington.edu/finlayso/ebook/algebraic/NR/NRmult.htm

• Journal of Mathematical Analysis and Application, Vol 5, No. 2, August 1996, Belgium.

BACK UP

Direct Matrix Solver

Definition

• The direct solution methods are characterized by their capacity to produce the solution vector for a given system of linear equations after a fixed number of operations.

Examples

• Gaussian Elimination

• Domain decomposition

• Thomas algorithm

• Tang’s algorithm

Applications

• 1D Rectangular or Radial Flow Problems (Thomas' Algorithm)

• Incompressible

• Slightly compressible

• 2D and 3D Flow Problems (Sparse Matrices)

Extra

Iterative Method

Definition

• In computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method.

Example

• Convergent Matrix

• Gauss-Seidel

• Newton-Raphson

• Jacobian Method

• Matrix Splitting

• Thomas Algorithm

• Etc.

Newton raphson • Imagine the following is the linear equation for the depth in

the reservoir,

More example to understand IM mathematically (flashback CM

memory) • Given a current approximation

x(k) = (x1(k), x2

(k), x3(k), …, xn

(k))

for x, the strategy of Jacobi's Method is to use the first equation and the current values of x2

(k), x3(k), …, xn

(k) to find a new value x1

(k+1), and similarly to find a new value xi(k) using

the i th equation and the old values of the other variables.

• This system can also be written as (decomposition)

• If we write D, L, and U for the diagonal, strict lower triangular and strict upper triangular and parts of A, respectively

• then Jacobi’s Method can be written in matrix-vector notation as

so that

Sparse Matrix

• In numerical analysis, a sparse matrix is a matrix in which most of the elements are zero. By contrast, if most of the elements are nonzero, then the matrix is considered dense. The fraction of zero elements (non-zero elements) in a matrix is called the sparsity(density).

• Conceptually, sparsity corresponds to systems which are loosely coupled. Consider a line of balls connected by springs from one to the next: this is a sparse system as only adjacent balls are coupled. By contrast, if the same line of balls had springs connecting each ball to all other balls, the system would correspond to a dense matrix. The concept of sparsity is useful in combinatorics and application areas such as network theory, which have a low density of significant data or connections.

• Large sparse matrices often appear in scientific or engineering applications when solving partial differential equations.