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This study proposes a method for designing stabilizing suboptimal controllers for nonlinear stochastic systems. These systems include time-invariant stochastic parameters that represent uncertainty of dynamics, posing two key difficulties…
In this paper we consider multi-dimensional partial differential equations of parabolic type involving divergence form operators that possess a discontinuous coefficient matrix along some smooth interface. The solution of the equation is…
We propose a novel numerical inversion algorithm for the coefficients of parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where…
This paper investigates oscillation-free stability conditions of numerical methods for linear parabolic partial differential equations with some example extrapolations to nonlinear equations. Not clearly understood, numerical oscillations…
We present compact semi-implicit finite difference schemes on structured grids for numerical solutions of the advection by an external velocity and by a speed in normal direction that are applicable in level set methods. The most involved…
High-precision numerical scheme for nonlinear hyperbolic evolution equations is proposed based on the spectral method. The detail discretization processes are discussed in case of one-dimensional Klein-Gordon equations. In conclusion, a…
Fourth-order accurate compact schemes for variable coefficient convection diffusion equations are considered. A sufficient condition for the stability of the fully discrete problem is derived using a difference equation based approach. The…
Boundary problem for linear partial differential algebraic equations system with multiple characteristic curves is considered. It is supposed that matrix-functions pencil of the system under consideration is smoothly equivalent to special…
We propose a second order finite volume scheme for nonlinear degenerate parabolic equations. For some of these models (porous media equation, drift-diffusion system for semiconductors, ...) it has been proved that the transient solution…
Various classes of stable finite difference schemes can be constructed to obtain a numerical solution. It is important to select among all stable schemes such a scheme that is optimal in terms of certain additional criteria. In this study,…
We consider locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of parabolic initial-boundary value problems with variable, possibly discontinuous in space…
Uncertainty quantification appears today as a crucial point in numerous branches of science and engineering. In the past two decades, a growing interest has been devoted to stochastic finite element method (SFEM) for the propagation of…
In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed, i.e., the solution…
This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including a quasi-Fermat theorem and a quasi-Mean Value Theorem. As an application, this paper develops several numerical…
We propose a high-order finite element method for linear fourth-order elliptic problems that is both nodally bound-preserving and mass-conservative, based on a variational inequality formulation. The method admits an equivalent strictly…
In this paper, we present a class of nonuniform time-stepping, high-order linear stabilized schemes that can preserve both the discrete energy stability and maximum-bound principle (MBP) for the time-fractional Allen-Cahn equation. To this…
The phase-field crystal equation, a parabolic, sixth-order and nonlinear partial differential equation, has generated considerable interest as a possible solution to problems arising in molecular dynamics. This is because the phase-field…
A three-point monotone difference scheme is proposed for solving a one-dimensional non-stationary convection-diffusion-reaction equation with variable coefficients. The scheme is based on a parabolic spline and allows to linearly reproduce…
We study necessary conditions for stability of a Numerov-type compact higher-order finite-difference scheme for the 1D homogeneous wave equation in the case of non-uniform spatial meshes. We first show that the uniform in time stability…
We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves…