Using Continuation to Make a Good Initial Guess
To solve a boundary value problem, you need to provide an initial guess for the solution. The quality of your initial guess can be critical to the solver performance, and to being able to solve the problem at all. However, coming up with a sufficiently good guess can be the most challenging part of solving a boundary value problem. Certainly, you should apply the knowledge of the problem's physical origin. Often a problem can be solved as a sequence of relatively simpler problems, i.e., a continuation. This section provides examples that illustrate how to use continuation to:
Example: Using Continuation to Solve a Difficult BVP
This example solves the differential equation
for , on the interval [-1 1], with boundary conditions and . For , the solution has a transition layer at . Because of this rapid change in the solution for small values of , the problem becomes difficult to solve numerically.
The example solves the problem as a sequence of relatively simpler problems, i.e., a continuation. The solution of one problem is used as the initial guess for solving the next problem.
The demo |
|Note This problem appears in  to illustrate the mesh selection capability of a well established BVP code COLSYS.|
bvp4ccan use. Because there is an additional known parameter , the functions must be of the form
shockBC. Note that
shockODEis vectorized to improve solver performance. The additional parameter is represented by
The example passes
e as an additional input argument to
bvp4c then passes this argument to the functions
shockBC when it evaluates them. See Additional BVP Solver Arguments for more information.
bvp4cpasses the additional argument to all the functions the user supplies.
bvp4c to use these functions to evaluate the partial derivatives by setting the options FJacobian and BCJacobian. Also set
'on' to indicate that the differential equation function
shockODE is vectorized.
bvp4cwith a guess structure that contains an initial mesh and a guess for values of the solution at the mesh points. A constant guess of and , and a mesh of five equally spaced points on [-1 1] suffice to solve the problem for . Use
bvpinitto form the guess structure.
bvp4cdoes not perform continuation automatically, but the code's user interface has been designed to make continuation easy. The code uses the output
bvp4cproduces for one value of
eas the guess in the next iteration.
Example: Using Continuation to Verify a Solution's Consistent Behavior
Falkner-Skan BVPs arise from similarity solutions of viscous, incompressible, laminar flow over a flat plate. An example is
for on the interval with boundary conditions , , and .
The BVP cannot be solved on an infinite interval, and it would be impractical to solve it for even a very large finite interval. So, the example tries to solve a sequence of problems posed on increasingly larger intervals to verify the solution's consistent behavior as the boundary approaches .
The example imposes the infinite boundary condition at a finite point called
infinity. The example then uses continuation in this end point to get convergence for increasingly larger values of
infinity. It uses
bvpinit to extrapolate the solution
sol for one value of
infinity as an initial guess for the new value of
infinity. The plot of each successive solution is superimposed over those of previous solutions so they can easily be compared for consistency.
The demo |
bvp4cwith a guess structure that contains an initial mesh and a guess for values of the solution at the mesh points. A crude mesh of five points and a constant guess that satisfies the boundary conditions are good enough to get convergence when
infinity = 3.
infinity = 3. It then prints the computed value of for comparison with the value reported by Cebeci and Keller .
infinity = 4,
6. It uses
bvpinitto extrapolate the solution
solfor one value of
infinityas an initial guess for the next value of
infinity. For each iteration, the example prints the computed value of and superimposes a plot of the solution in the existing figure.
Note that the values approach 0.92768 as reported by Cebeci and Keller. The superimposed plots confirm the consistency of the solution's behavior.
|Solving BVP Problems||Solving Singular BVPs|