Interface, Free and Moving Boundary Value Problems
Free Boundary Value Problems for ODEs
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
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
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
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.
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.
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Email: rfazio@dipmat.unime.it