Interface, Free and Moving Boundary Value Problems

Free Boundary Value Problems for ODEs

- Similarity and Numerical Analysis for Free Boundary Value Problems
- Normal Variables Transformation Method Applied to Free Boundary Value Problems
- Numerical Transformation Methods: A Constructive Approach
- A Numerical Test for the Existence and Uniqueness of Solution of Free Boundary Problems
- A Similarity Approach to the Numerical Solution of Free Boundary Problems
- A Non-Iterative Transformation Method for Newton's Free Boundary Problem

** Abstract:**
We consider the similarity properties of nonlinear ordinary free
boundary value problems governed by a second order differential
equation.
By requiring the invariance of the differential equation with
respect to a stretching and a spiral group we characterize
two classes of problems that can be solved by a non-iterative
initial value method.
Moreover, an important extension of the method is introduced by
requiring that some parameter is transformed by the
stretching group.

** Abstract:**
We extend the application of group analysis approach to determining the
numerical solution of free boundary value problems.
If the differential problem is invariant under a translation group of
transformations we will formulate
a non-iterative method of solution.
This is done by introducing the concept of normal variables.
Application of the method to two problems in the class characterized
produces correct numerical results.
Moreover, introducing a parameter into the differential problem
and requiring invariance under an extended stretching group
we give an iterative method applicable to any free boundary
value problem.

** Abstract:**
Numerical transformation methods are initial value methods for the
solution of boundary value problems. Usually these methods have been
considered as non-iterative but ad hoc; their applicability depending
upon some invariance properties of the boundary value problem under
investigation.
In this paper a constructive approach in order to define two numerical
transformation methods was considered. As a main result we proved an
unification theorem for the underlining theory. In this way we
highlighted a widely applicable iterative transformation method.

** Abstract:**
The aim of this work is to introduce a numerical test
for the existence and uniqueness of solution of free boundary problems
governed by an ordinary differential equation.
The main result is given by a theorem relating the
existence and uniqueness question to the number of real zeros of a
function implicitly defined within the formulation of the iterative
transformation method. As a consequence, we can investigate
the existence and uniqueness of solution
by studying the behaviour of that function.
Within such a context the numerical test is illustrated by two
examples.

** Abstract:**
The aim of this work is to point out that within a similarity approach
some classes of free boundary value problems
governed by ordinary differential equations
can be transformed to initial value problems.
The interest in the numerical solution of free boundary problems
arises because these are always nonlinear problems. Furthermore we show
that free boundary problems arise also via a similarity analysis of
moving boundary hyperbolic problems and they can be
obtained as approximation of boundary value problems defined
on infinite intervals.
Most of the theoretical content of this survey is original: it
generalizes
and unifies results already available in literature.
As far as applications of the proposed approach are concerned,
three problems of interest are considered and numerical results
for each of them are reported.

** Abstract:**
In book II of Newton's Principia Mathematica of 1687 several applicative problems are introduced and solved.
There, we can find the formulation of the first calculus of variations problem that leads to the first free boundary problem of history.
The general calculus of variations problem is concerned with the optimal shape design for the motion of projectiles subject to air resistance.
Here, for Newton's optimal nose cone free boundary problem, we define a non-iterative initial value method which is referred in literature as transformation method.
To define this method we apply invariance properties of Newton's free boundary problem under a scaling group of point transformations.
Finally, we compare our non-iterative numerical results with those available in literature and obtained via an iterative shooting method.
We emphasize that our non-iterative method is faster than shooting or collocation methods and does not need any preliminary computation to test the target function as the iterative method or even provide any initial iterate.
Moreover, applying Buckingham Pi-Theorem we get the functional relation between the unknown free boundary and the nose cone radius and height.

Free Boundary Formulation for BVPs on Infinite Intervals

Selected List of Papers- The Blasius Problem Formulated as a Free Boundary Value Problem
- The Falkner-Skan Equation: Numerical Solutions within Group Invariance Theory
- A Novel Approach to the Numerical Solution of Boundary Value Problems on Infinite Intervals
- A Survey on Free Boundary Identification of the Truncated Boundary in Numerical BVPs on Infinite Intervals
- A Free Boundary Approach and the Box Scheme for Boundary Value Problems on Infinite Intervals
- On the moving boundary formulation for parabolic problems on unbounded domains

** Abstract:** In the present paper we
point out that the correct way to solve the Blasius problem by
numerical means is to reformulate it as a free boundary value problem.
In the new formulation the truncated boundary (instead of infinity) is
the unknown free boundary and it has to be determined as part of the
numerical solution. Taking into account the "partial" invariance of the
mathematical model at hand with respect to a stretching group we define
a non-iterative transformation method. Further, we compare the improved
numerical results, obtained by the method in point, with analytical and
numerical ones. Moreover, the numerical results confirm that the
question of accuracy depends on the value of the free boundary.
Therefore, this indicates that boundary value problems with a boundary
condition at infinity should be numerically reformulated as free
boundary value problems.

** Abstract:** The iterative transformation
method, defined within the framework of the group invariance theory, is
applied to the numerical solution of the Falkner-Skan equation with
relevant boundary conditions. In this problem a boundary condition at
infinity is imposed which is not suitable for a numerical use. In order
to overcome this difficulty we introduce a free boundary formulation of
the problem, and we define the iterative transformation method that
reduces the free boundary formulation to a sequences of initial value
problems. Moreover, as far as the value of the wall shear stress is
concerned we propose a numerical test of convergence. The usefulness of
our approach is illustrated by considering the wall shear stress for
the classical Homann and Hiemenz flows. In the Homann's case we apply
the proposed numerical test of convergence, and meaningful numerical
results are listed. Moreover, for both cases we compare our results
with those reported in literature.

** Abstract:** The classical numerical
treatment of two-point boundary value problems on infinite intervals is
based on the introduction of a truncated boundary (instead of infinity)
where appropriate boundary conditions are imposed. Then, the truncated
boundary allowing for a satisfactory accuracy is computed by trial.
Motivated by several problems of interest in boundary layer theory,
here we consider boundary value problems on infinite intervals governed
by a third order ordinary differential equation. We highlight a novel
approach in order to define the truncated boundary. The main result is
the convergence of the solution of our formulation to the solution of
the original problem as a suitable parameter goes to zero. In the
proposed formulation the truncated boundary is an unknown free boundary
and has to be determined as part of the solution. For the numerical
solution of the free boundary formulation a non-iterative and an
iterative transformation method are introduced. Furthermore, we
characterize the class of free boundary value problems that can be
solved non-iteratively. A nonlinear flow problem involving two physical
parameters and belonging to the characterized class of problems is then
solved. Moreover, the Falkner-Skan equation with relevant boundary
conditions is considered and representative results, obtained by the
iterative transformation method, are listed for the Homann flow. All
the obtained numerical results clearly indicate the effectiveness of
the proposed approach. Finally, we discuss some open problems and
possible extensions for our approach.

** Abstract:**
A free boundary formulation for the
numerical solution of boundary value problems on infinite
intervals was proposed recently in [R. Fazio,
SIAM J. Numer. Anal., 33 (1996) 1473-1483].
We consider here a survey on recent developments related to the
free boundary identification of the truncated boundary.
The goals of this survey are: to recall the reasoning for a
free boundary identification of the truncated boundary,
to report on a comparison of numerical
results obtained for a classical test problem
by three approaches available in the literature,
and to propose some possible ways to extend the free boundary
approach to the numerical solution of problems defined on the
whole real line.

** Abstract:** A free boundary approach for
the numerical solution
of boundary value problems governed by a third order differential
equation and defined on infinite intervals was proposed recently in [R.
Fazio, SIAM J. Numer. Anal., 33 (1996), pp. 1473-1483].
In this paper we extend that approach to problems governed by
a system of first order differential equations.
For the numerical solution of the free boundary problem
the box difference scheme is considered.
To show the validity of the free boundary approach under concern, a
nonlinear model of interest in foundation engineering
is solved.

** Abstract:**
The aim of this paper is to propose an original numerical approach for
parabolic problems whose governing
equations are defined on unbounded domains.We are interested in
studying the class of problems admitting
invariance property to Lie group of scalings. Thanks to similarity
analysis the parabolic problem can be
transformed into an equivalent boundary value problem governed by an
ordinary differential equation and
defined on an infinite interval. A free boundary formulation and a
convergence theorem for this kind of
transformed problems are available in [R. Fazio, A novel approach to
the numerical solution of boundary
value problems on infinite intervals, SIAM J. Numer. Anal. 33 (1996),
pp. 1473–1483]. Depending on
its scaling invariance properties, the free boundary problem is then
solved numerically using either a
noniterative, or an iterative method. Finally, the solution of the
parabolic problem is retrieved by applying
the inverse map of similarity.

Moving Boundary Parabolic Problems

- The Iterative Transformation Method: Numerical Solution of One-Dimensional Parabolic Moving Boundary Problems
- Similarity Analysis for Moving Boundary Parabolic Problems

** Abstract:**
The main contribution of this paper is the application of the
iterative transformation method to the numerical solution of the
sequence of free boundary problems obtained from one-dimensional
parabolic moving boundary problems via the implicit Euler's
method. The combination of the two methods represents a numerical
approach to the solution of those problems. Three parabolic moving
boundary problems, two with explicit and one with implicit moving
boundary conditions, are solved in order to test the validity of
the proposed approach. As far as the moving boundary position is
concerned the obtained numerical results are found to be in
agreement with those available in literature.

** Abstract:** The reduction of
one-dimensional moving boundary parabolic problems to free
boundary problems governed by ordinary differential equations is
considered.
We indicate the iterative transformation method, defined within the
theory of group invariance,
as an effective way to solve the transformed problems by numerical
means.
Then we solve two problems of interest. The numerical results are found
in agreement with exact or approximate one.

Moving Boundary Hyperbolic Problems

- A Nonlinear Hyperbolic Free Boundary Value Problem
- A Moving Boundary Hyperbolic Problem for a Stress Impact in a Bar of Rate-type Material
- An Implicit Difference Scheme for a Moving Boundary Hyperbolic Problem
- Similarity and Numerical Analysis of a Singular Moving Boundary Hyperbolic Problem

** Abstract:** The present paper is
concerned with the application of a non-iterative transformation method
to the numerical solution of a nonlinear hyperbolic free boundary value
problem. Making use of the similarity analysis approach to the
hyperbolic model describing time dependent velocity impact to nonlinear
inhomogeneous thin rods we recover a free boundary value problem. Since
exact solutions are known only in some particular cases, we consider
application of numerical methods of integration. Usually iterative
numerical methods of solution are known to be applicable to free
boundary value problems. However, we can prove that the ordinary
differential equation related to the model in point is invariant with
respect to a stretching group of transformations. This is the hint to
apply group properties and to deduce an ad hoc non-iterative
transformation method.

** Abstract:** In this paper we present
some results obtained by studying the mathematical model In this paper
we present some results obtained by studying the mathematical model
describing a moving boundary hyperbolic problem related to a time
dependent stress impact in a bar of Maxwell-like material. Due to the
impact a shock front propagates with a finite speed. Here our interest
is to underline the influence of the dissipative term on the
propagation of the shock front. In the framework of the similarity
analysis we are able to reduce the moving boundary hyperbolic problem
to a free boundary value problem for an ordinary differential system.
It is then possible, by applying two numerical transformation methods,
to solve the free boundary value problem numerically. The influence of
the dissipative term is evident: the free boundary (that defines the
shock front propagation) is an increasing function of the dissipative
coefficient.

** Abstract:** In this paper an implicit
difference scheme is defined for a moving boundary hyperbolic problem,
which describes a shock front propagation in a constant state. We have
reformulated the problem to a fixed boundary domain where an implicit
difference scheme is proposed. As is well known, the equivalent
condition for the convergence of a consistent scheme is its stability.
However, the only reliable methods of stability analysis are based on
linear theory. Moreover, the pertinent literature provides simple
examples where the linearization of a nonlinear scheme leads to
incorrect stability results. On an experimental basis a discrete
perturbation stability analysis was then considered. In order to
investigate the convergence of the scheme we considered a particular
example where an approximate similarity solution is known. In this
case, we point out the numerical convergence of the scheme. Even more
important is that a possible way to assess the numerical accuracy when
the similarity solution does not exist is suggested.

** Abstract:**
In [SIAM Rev., 40 (1998) 616--635], we emphasized the relevance of a
combination of similarity and numerical
analysis for the numerical solution of moving boundary
hyperbolic problems. Here we report on results obtained for one problem
of the above class that is singular at
the moving boundary.

- A Lagrangian Central Scheme for Multi-Fluid Flows
- Comparison of Two Conservative Schemes for Hyperbolic Interface Problems.
- Moving Mesh Methods for Hyperbolic Conservation Laws Using CLAWPACK
- A Lagrangian Central Schemes and Second Order Boundary Conditions for 1D Interface and Piston Problems

** Abstract:**
We develop a central scheme for multi-fluid
flows in Lagrangian coordinates. The main contribution is the
derivation of a special
equation of state to be imposed at the interface in order to avoid
non-physical oscillations. The proposed scheme is validated by
solving several tests concerning one-dimensional hyperbolic
interface problems.

** Abstract:**
We describe two conservative schemes recently proposed for the numerical solution of hyperbolic inteface problems,
and compare the two schemes on a piston problem and a shock tube problem.

** Abstract:**
The wave propagation method, implemented in the CLAWPACK software
package, is a
high-resolution finite-volume method for nonlinear conservation laws
and other
hyperbolic systems.
Here we show how this method can be easily applied to one-dimensional
conservation
laws on moving grids. The method is tested on a shock-tube problem with
multiple reflections and on a
more interesting problem with a moving piston bounding a tube
containing two gases.
The moving grid is used to track both the interface and the piston
motion.

** Abstract:**
We study high-resolution Lagrangian central schemes for the
one-dimensional
system of conservation laws describing the evolution of two gases in
slab geometry
separated by an interface. By using Lagrangian coordinates, the
interface is transformed
to a fixed coordinate in the computational domain and, as a
consequence,
the movement of the interface is obtained as a byproduct of the
numerical solution.
The main contribution is the derivation of a special equation of state
to be imposed
at the interface in order to avoid non-physical oscillations. Suitable
boundary conditions
at the piston that guarantee second order convergence are described.
We compare the solution of the piston problem to other results
available in the
literature and to a reference solution obtained in the adiabatic
approximation. A
shock-interface interaction problem is also treated. The results on
these tests are in
good agreement with those obtained by other methods.

Address:

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