Tchavdar T. Marinov

NOVEL NUMERICAL APPROACH TO SOLITARY{WAVE SOLUTIONS IDENTIFICATION OF BOUSSINESQ AND KORTEWEG{DE VRIES EQUATIONS

A special numerical technique has been developed for identification of solitary wave solutions of Boussinesq and Korteweg–de Vries equations. Stationary localized waves are considered in the frame moving to the right. The original ill-posed problem is transferred into a problem of the unknown coefficient from over-posed boundary data in which the trivial solution is excluded. The Method of Variational Imbedding is used for solving the inverse problem. The generalized sixth-order Boussinesq equation is considered for illustrations.

A Fully Coupled Solver for Incompressible Navier–Stokes Equations using Operator Splitting

The steady incompressible Navier–Stokes equations are coupled by a Poisson equation for the pressure from which the continuity equation is subtracted. The equivalence to the original N–S problem is proved. Fictitious time is added and vectorial operator-splitting is employed leaving the system coupled at each fractional-time step which allows satisfaction of the boundary conditions without introducing artificial conditions for the pressure. Conservative second-order approximations for the convective terms are employed on a staggered grid. The splitting algorithm for the 3D case is verified through an analytic solution test. The stability of the method at high values of Reynolds number is illustrated by accurate numerical solutions for the flow in a lid-driven rectangular cavity with aspect ratio 1 and 2, as well as for the flow after a back-facing step.

Inverse problem for coefficient identification in the Euler-Bernoulli equation

A special technique has been developed for the identification of the solution and the unknown coefficient in the Euler-Bernoulli equation. The original problem of the unknown coefficient identification from over-posed data is transferred into a higher-order well-posed problem following the idea of the Method of Variational Imbedding. The new boundary value problem is solved by means of an iterative finite difference scheme. The scheme is thoroughly validated and is shown to have a second-order truncation error.

Identification of solitary-wave solutions as an inverse problem: Application to shapes with oscillatory tails

The propagation of stationary solitary waves on an infinite elastic rod on elastic foundation equation is considered. The asymptotic boundary conditions admit the trivial solution along with the solution of type of solitary wave, which is a bifurcation problem. The bifurcation is treated by prescribing the solution in the origin and introducing an unknown coefficient in the equation. Making use of the method of variational imbedding, the inverse problem for the coefficient identification is reformulated as a higher-order boundary value problem. The latter is solved by means of an iterative difference scheme, which is thoroughly validated. Solitary waves with oscillatory tails are obtained for different values of tension and linear restoring force. Special attention is devoted to the case with negative tension, when the solutions have oscillatory tails.

Coefficient identification in Euler-Bernoulli equation from over-posed data

A method for solving the inverse problem for coefficient identification in the Euler-Bernoulli equation from over-posed data is presented. The original inverse problem is replaced by a minimization problem. The method is applied to the problem for identifying the coefficient in the case when it is a piece-wise polynomial function. Several examples are elaborated and the numerical results confirm that the solution of the imbedding problem coincides with the direct simulation of the original problem within the second order of approximation.

Characteristic Line Based Schemes for Solving a Quasilinear Hierarchical Size-Structured Model

New schemes, based on the characteristics line method, for solving a hierarchical size-structured model with nonlinear growth, mortality and reproduction rate are developed. The idea of the schemes is not to follow the characteristics from the initial condition, but for each time-step to find the origins of the grid nodes at the previous time level. Numerical tests, including comparison with exact solutions for the new schemes, are elaborated. Numerical results that confirm the theoretical order of convergence of the new schemes are presented.

Inverse problem for coefficient identification in SIR epidemic models

Abstract This work deals with the development of a numerical method for solving an inverse problem for identifying coefficients from over-posed data in an SIR mathematical model of infectious diseases spread through a population. The parameters are identified using a generalized Least Squares Method which is similar to the technique called Method of Variational Imbedding, where the original inverse problem is replaced by a minimization problem. A difference scheme and a numerical algorithm for solving the parameter identification problem are developed. The correctness of the embedded problem is discussed. Numerical results of the parameters, representing the solution to the inverse problem, are presented.

Method of Variational Imbedding for the Inverse Problem of Boundary-Layer Thickness Identification

The inverse problem of identification of boundary-layer thickness is replaced by the higher-order boundary value problem for the Euler–Lagrange equations for minimization of the quadratic functional of the original system (Method of Variational Imbedding – MVI). The imbedding problem is correct in the sense of Hadamard and consists of an explicit differential equation for the boundary-layer thickness. The existence and uniqueness of solution of the linearized imbedding problem is demonstrated and a difference scheme of splitting type is proposed for its numerical solution. The performance of the technique is demonstrated for three different boundary-layer problems: the Blasius problem, flow in the vicinity of plane stagnation point and the flow in the leading stagnation point on a circular cylinder. Comparisons with the self-similar solutions where available are quantitatively very good.

Inverse problem for coefficient identification in euler-bernoulli equation by linear spline approximation

We display the performance of the technique called Method of Variational Imbedding for solving the inverse problem of coefficient identification in Euler-Bernoulli equation from over-posed data The original inverse problem is replaced by a minimization problem The Euler-Lagrange equations comprise an eight-order equation for the solution of the original equation and an explicit system of equations for the coefficients of the spline approximation of the unknown coefficient Featuring examples are elaborated numerically The numerical results confirm that the solution of the imbedded problem coincides with the exact solution of the original problem within the order of approximation error.

Conservation properties of vectorial operator splitting

This work is concerned with the conservation properties of a new vectorial operator splitting scheme for solving the incompressible Navier-Stokes equations. It is proven that the difference approximation of the advection operator conserves square of velocity components and the kinetic energy as the differential operator does, while pressure term conserves only the kinetic energy. Some analytical requirements necessary to be satisfied of difference schemes for incompressible Navier-Stokes equations are formulated and discussed. The properties of the methods are illustrated with results from numerical computations for lid-driven cavity flow.