Matlab code for keller box method: Applications to fluid flow and heat transfer problems
quileanistbestMatlab code for keller box method: A numerical solution for ordinary and partial differential equations
The Keller box method is a powerful and popular technique for solving nonlinear algebraic equations, ordinary differential equations (ODEs), and partial differential equations (PDEs). It is based on the Newton-Raphson method, which iteratively finds the roots of an equation by linearizing it around an initial guess. The Keller box method extends this idea to systems of equations, by using a set of parabolic subequations to approximate the original system.
Matlab Code For Keller Box Method
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In this article, we will explain how the Keller box method works, and how to implement it in Matlab. We will also show some examples of applying the method to boundary layer problems in fluid mechanics, which are often described by coupled ODEs or PDEs.
How the Keller box method works
The Keller box method can be derived by first writing the system of equations as:
f(x) = 0
where x is a vector of unknown variables, and f is a vector of functions. The goal is to find x such that f(x) is zero.
The Newton-Raphson method starts with an initial guess x0, and then updates it according to:
xn+1 = xn - J(xn)f(xn)
where J(xn) is the Jacobian matrix of f, which contains the partial derivatives of f with respect to x. The inverse of the Jacobian matrix is used to linearize the system around xn.
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The Keller box method modifies this formula by introducing a set of parabolic subequations:
y''(t) = f(y(t))
where y(t) is a vector of auxiliary variables, and t is a parameter that ranges from 0 to 1. The boundary conditions for these subequations are:
y(0) = xn
y(1) = xn+1
The idea is to use these subequations to approximate the original system, by finding y(t) such that f(y(t)) = 0 for all t. This can be done by using a finite difference scheme, such as:
y''(ti) = (y(ti+1) - 2y(ti) + y(ti-1))/h
where h = 1/N, and N is the number of grid points. The boundary conditions can be incorporated by using ghost points, such as:
y(-h) = 2xn - y(h)
y(1+h) = 2xn+1 - y(1-h)
The resulting system of equations can be written as:
Ay = b
where A is a tridiagonal matrix, y is a vector of unknowns, and b is a vector of constants. This system can be solved by using a direct or iterative method, such as Gaussian elimination or Gauss-Seidel.
How to implement the Keller box method in Matlab
To implement the Keller box method in Matlab, we need to define the following functions:f(x): This function evaluates the system of equations at a given point x.J(x): This function evaluates the Jacobian matrix of f at a given point x.
Examples of Keller box method for boundary layer problems
Boundary layer problems are common in fluid mechanics, where the flow near a solid surface is influenced by viscous effects. These problems often involve coupled ODEs or PDEs that describe the velocity, temperature, and concentration profiles of the fluid. The Keller box method can be used to solve these problems by transforming them into a system of nonlinear algebraic equations.
One example of a boundary layer problem is the Blasius equation, which models the laminar flow over a flat plate. The Blasius equation is given by:
f'''(eta) + 0.5f(eta)f''(eta) = 0
where f(eta) is the dimensionless stream function, and eta is the similarity variable. The boundary conditions are:
f(0) = f'(0) = 0
f'(inf) = 1
The Keller box method can be applied to this problem by introducing two auxiliary variables g(eta) and h(eta), such that:
f'(eta) = g(eta)
f''(eta) = h(eta)
The original equation and boundary conditions can then be written as:
h'(eta) + 0.5f(eta)h(eta) = 0
f(0) = g(0) = 0
g(inf) = 1
The Keller box method can then be implemented in Matlab by using a finite difference scheme to discretize the equations and boundary conditions, and then solving the resulting system of equations by using a direct or iterative method.
Advantages and disadvantages of Keller box method
The Keller box method has some advantages and disadvantages compared to other numerical methods for solving nonlinear algebraic equations, ODEs, and PDEs. Some of the advantages are:The method is simple and easy to implement in Matlab.The method can handle systems of equations with any number of variables.The method can deal with singularities and turning points in the solutions.The method can handle boundary conditions at infinity by using ghost points.The method can achieve high accuracy with relatively few grid points.
Some of the disadvantages are:The method requires an initial guess for each variable at each grid point.The method may not converge if the initial guess is too far from the true solution.The method may be sensitive to the choice of grid spacing and parabolic parameter.The method may not preserve some properties of the original equations, such as symmetry or conservation laws.The method may not be efficient for large-scale problems or problems with complex geometries.