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The current status of numerical simulations of galaxy formation is reviewed. After a short description of the main numerical simulation techniques, three sample applications illustrate how numerical simulations have provided deeper insight…
We study one-dimensional viscoelastic phase transitions modeled by a Ginzburg--Landau energy with a non-convex cubic stress-strain law. Extending the isothermal model, we couple the momentum equation to a heat equation for the temperature…
In this paper we study Cauchy problem for thermoelastic plate equations with friction or structural damping in $\mathbb{R}^n$, $n\geq1$, where the heat conduction is modeled by Fourier's law. We explain some qualitative properties of…
Three-dimensional nanoarchitectures are widely used across various areas of physics, including spintronics, photonics, and superconductivity. In this regard, thin curved 3D membranes are especially interesting for applications in nano- and…
In this paper we propose a nonconforming finite element method for the solution of the ill-posed elliptic Cauchy problem. We prove error estimates using continuous dependence estimates in the $L^2$-norm. The effect of perturbations in data…
A numerical algorithm for mean curvature flow of closed mean convex surfaces with surgery is proposed. The method uses a finite element based mean curvature flow algorithm based on a coupled partial differential equation system which…
We analyse an algorithm of transition between Cauchy problems for second-order wave equations and first-order symmetric hyperbolic systems in case the coefficients as well as the data are non-smooth, even allowing for regularity below the…
We consider numerical methods for linear parabolic equations in one spatial dimension having piecewise constant diffusion coefficients defined by a one parameter family of interface conditions at the discontinuity. We construct immersed…
Computational technologies for the approximate solution of multidimensional boundary value problems often rely on irregular computational meshes and finite-volume approximations. In this framework, the discrete problem represents the…
This paper concerns the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space $\mathbb{R}^{2}$ with vacuum as far field density. It is proved that the 2D nonhomogeneous…
In the present chapter we focus on the fundamentals of non-grid-conforming numerical approaches to simulating particulate flows, implementation issues and grid convergence vs. available reference data. The main idea is to avoid adapting the…
In the study of gas dynamics, theoretical modeling and numerical simulation are mostly set up with deterministic settings. Given the coarse-grained modeling in theories of fluids, considerable uncertainties may exist between flow-field…
Slug flows are a typical intermittent two-phase flow pattern that can occur in submarine pipelines connecting the wells to the production facility and that is known to cause undesired consequences. In this context, computational fluid…
We study the problem to provide a triangular form based on implicit differential equations for non-linear multi-input systems with respect to the flatness property. Furthermore, we suggest a constructive method for the transformation of a…
We perform full-scale numerical simulation of instability of weakly nonlinear waves on the surface of deep fluid. We show that the instability development leads to chaotization and formation of wave turbulence. We study instability both of…
We devise a new time-stepping algorithm for two-dimensional nonlinear unsteady surface and interfacial waves. The algorithm uses Cauchy's integral formula, which only requires information on the interface, to solve Laplace equation by using…
In this paper, a stabilized extended finite element method is proposed for Stokes interface problems on unfitted triangulation elements which do not require the interface align with the triangulation. The velocity solution and pressure…
We study the Cauchy problem for the cubic nonlinear Schroedinger equation, perturbed by (higher order) dissipative nonlinearities. We prove global in-time existence of solutions for general initial data in the energy space. In particular we…
We study the Cauchy problem of three-dimensional compressible non-isentropic magnetohydrodynamic (MHD) fluids with both interior and far field vacuum states. Applying delicate energy estimates, initial layer analysis, and continuation…
Despite its numerical challenges, finite element method is used to compute viscous fluid flow. A consensus on the cause of numerical problems has been reached; however, general algorithms---allowing a robust and accurate simulation for any…