Numerical computation of transonic flows by finite-element and finite-difference methods

Studies on applications of the finite element approach to transonic flow calculations are reported. Different discretization techniques of the differential equations and boundary conditions are compared. Finite element analogs of Murman's mixed type finite difference operators for small disturbance formulations were constructed and the time dependent approach (using finite differences in time and finite elements in space) was examined

A survey on stationary problems, Green's functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions

In this paper, we present a survey of recent results on the Green's functions and on spectrum for stationary problems with nonlocal boundary conditions. Results of Lithuanian mathematicians in the field of differential and numerical problems with nonlocal boundary conditions are described. *The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014)

A New Unconditionally Stable Method for Telegraph Equation Based on Associated Hermite Orthogonal Functions

The present paper proposes a new unconditionally stable method to solve telegraph equation by using associated Hermite (AH) orthogonal functions. Unlike other numerical approaches, the time variables in the given equation can be handled analytically by AH basis functions. By using the Galerkin’s method, one can eliminate the time variables from calculations, which results in a series of implicit equations. And the coefficients of results for all orders can then be obtained by the expanded equations and the numerical results can be reconstructed during the computing process. The precision and stability of the proposed method are proved by some examples, which show the numerical solution acquired is acceptable when compared with some existing methods

Explicit-in-Time Variational Formulations for Goal-Oriented Adaptivity

Goal-Oriented Adaptivity (GOA) is a powerful tool to accurately approximate physically relevant features of the solution of Partial Differential Equations (PDEs). It delivers optimal grids to solve challenging engineering problems. In time dependent problems, GOA requires to represent the error in the Quantity of Interest (QoI) as an integral over the whole space-time domain in order to reduce it via adaptive refinements. A full space-time variational formulation of the problem allows the aforementioned error representation. Thus, variational spacetime formulations for PDEs have been of great interest in the last decades, among other things, because they allow to develop mesh-adaptive algorithms. Since it is known that implicit time marching schemes have variational structure, they are often employed for GOA in time-domain problems. When coming to explicit-intime methods, these were introduced for Ordinary In this dissertation, we prove that the explicit Runge-Kutta (RK) methods can be expressed as discontinuous-in-time Petrov-Galerkin (dPG) methods for the linear advection-diffusion equation. We systematically build trial and test functions that, after exact integration in time, lead to one, two, and general stage explicit RK methods. This approach enables us to reproduce the existing time domain goal-oriented adaptive algorithms using explicit methods in time. Here, we employ the lowest order dPG formulation that we propose to recover the Forward Euler method and we derive an appropriate error representation. Then, we propose an explicit-in-time goal-oriented adaptive algorithm that performs local refinements in space. In terms of time domain adaptivity, we impose the Courant-Friedrichs-Lewy (CFL) condition to ensure the stability of the method. We provide some numerical results in one-dimensional (1D)+time for the diffusion and advection-diffusion equations to show the performance of the proposed algorithm. On the other hand, time-domain adaptive algorithms involve solving a dual problem that runs backwards in time. This process is, in general, computationally expensive in terms of memory storage. In this work, we dene a pseudo-dual problem that runs forwards in time. We also describe a forward-in-time adaptive algorithm that works for some specific problems. Although it is not possible to dene a general dual problem running forwards in time that provides information about future states, we provide numerical evidence via one-dimensional problems in space to illustrate the efficiency of our algorithm as well as its limitations. As a complementary method, we propose a hybrid algorithm that employs the classical backward-in-time dual problem once and then performs the adaptive process forwards in time. We also generalize a novel error representation for goal-oriented adaptivity using (unconventional) pseudo-dual problems in the context of frequency-domain wave-propagation problems to the time-dependent wave equation. We show via 1D+time numerical results that the upper bounds for the new error representation are sharper than the classical ones. Therefore, this new error representation can be used to design more efficient goal-oriented adaptive methodologies. Finally, as classical Galerkin methods may lead to instabilities in advection-dominated-diffusion problems and therefore, inappropriate refinements, we propose a novel stabilized discretization method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. This method combines the benefits resulting from Isogeometric Analysis (IGA), residual minimization, and Alternating Direction Implicit (ADI) methods. We employ second order ADI time integrator schemes, B-spline basis functions in space and, at each time step, we solve a stabilized mixed method based on residual minimization. We show that the resulting system of linear equations has a Kronecker product structure, which results in a linear computational cost of the direct solver, even using implicit time integration schemes together with the stabilized mixed formulation. We test our method in 2D and 3D+time advection-diffusion problems. The derivation of a time-domain goal-oriented strategy based on iGRM will be considered in future works