Parabolic Interface Research Articles

Short-time height distribution in the one-dimensional Kardar-Parisi-Zhang equation: Starting from a parabola.

We study the probability distribution P(H,t,L) of the surface height h(x=0,t)=H in the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension when starting from a parabolic interface, h(x,t=0)=x^{2}/L. The limits of L→∞ and L→0 have been recently solved exactly for any t>0. Here we address the early-time behavior of P(H,t,L) for general L. We employ the weak-noise theory-a variant of WKB approximation-which yields the optimal history of the interface, conditioned on reaching the given height H at the origin at time t. We find that at small HP(H,t,L) is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as -lnP=f_{+}|H|^{5/2}/t^{1/2} and f_{-}|H|^{3/2}/t^{1/2}. The factor f_{+}(L,t) monotonically increases as a function of L, interpolating between time-independent values at L=0 and L=∞ that were previously known. The factor f_{-} is independent of L and t, signaling universality of this tail for a whole class of deterministic initial conditions.

A Fictitious Domain Method with Distributed Lagrange Multiplier for Parabolic Problems With Moving Interfaces

In this paper, we study the fictitious domain method with distributed Lagrange multiplier for the jump-coefficient parabolic problems with moving interfaces. The equivalence between the fictitious domain weak form and the standard weak form of a parabolic interface problem is proved, and the uniform well-posedness of the full discretization of fictitious domain finite element method with distributed Lagrange multiplier is demonstrated. We further analyze the convergence properties for the fully discrete finite element approximation in the norms of $$L^2$$ , $$H^1$$ and a new energy norm. On the other hand, we introduce a subgrid integration technique in order to allow the fictitious domain finite element method to be performed on the triangular meshes without doing any interpolation between the authentic domain and the fictitious domain. Numerical experiments confirm the theoretical results, and show the good performances of the proposed schemes.

Convergence of a Second-Order Linearized BDF–IPDG for Nonlinear Parabolic Equations with Discontinuous Coefficients

We study the anti-symmetric interior over-penalized discontinuous Galerkin finite element methods for solving nonlinear parabolic interface problems with second-order backward difference formula for the time discretization, where the diffusion coefficient depends on the unknown solution and is discontinuous across the interface. We present optimal-order error estimates for the finite element solution based on piecewise regularity of the solution.

A Posteriori Error Analysis of Two-Step Backward Differentiation Formula Finite Element Approximation for Parabolic Interface Problems

This paper studies a residual-based a posteriori error estimates for linear parabolic interface problems in a bounded convex polygonal domain in $$\mathbb {R}^2$$R2. We use the standard linear finite element spaces in space which are allowed to change in time and the two-step backward differentiation formula (BDF-2) approximation at equidistant time step is used for the time discretizations. The essential ingredients in the error analysis are the continuous piecewise quadratic space---time BDF-2 reconstruction and Scott---Zhang interpolation estimates. Optimal order in time and an almost optimal order in space error estimates are derived in the $$L^{\infty }(L^{2})$$Lź(L2)-norm using only energy method. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical experiments are performed to validate the asymptotic behaviour of the derived error estimators.

Finite difference approximation for parabolic interface problem with time-dependent coefficients

The convergence of difference scheme for two-dimensional initial boundary value problem for the heat equation with concentrated capacity and time-dependent coefficients of the space derivatives, is considered. An estimate of the rate of convergence in a special discrete W~12,1/2 Sobolev norm, compatible with the smoothness of the coefficients and solution, is proved.

Discontinuous Galerkin immersed finite element methods for parabolic interface problems

In this article, interior penalty discontinuous Galerkin methods using immersed finite element functions are employed to solve parabolic interface problems. Typical semi-discrete and fully discrete schemes are presented and analyzed. Optimal convergence for both semi-discrete and fully discrete schemes is proved. Some numerical experiments are provided to validate our theoretical results.

P 1-nonconforming triangular finite element method for elliptic and parabolic interface problems

The lowest order P1-nonconforming triangular finite element method (FEM) for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optimal order error estimates are obtained in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.

Advances of finite element analysis for elliptic, parabolic and Maxwell interface equations

The development of finite element methods for interface problems in the recent two decades is reviewed with an emphasis on elliptic, parabolic and Maxwell interface problems. We summarize some important results, e.g., regularity, variational forms, discretization schemes, numerical quadrature, interface approxima-fition, and convergence analysis. The state of the art for these problems is described along with the history of developments. Finally, we outline some applications governed by the PDE systems with interfaces.

Local discontinuous Galerkin method for parabolic interface problems

In this paper, the minimal dissipation local discontinuous Galerkin method is studied to solve the parabolic interface problems in two-dimensional convex polygonal domains. The interface may be arbitrary smooth curves. The proposed method is proved to be L 2 stable and the order of error estimates in the given norm is O(h|logh|1/2). Numerical experiments show the efficiency and accuracy of the method.

Convergence of a linearized second-order BDF-FEM for nonlinear parabolic interface problems

We present an almost optimal error estimate for the finite element solution of a nonlinear parabolic interface problem, where the coefficient depends on the unknown variable and is discontinuous along an interface inside the computational domain. A linearized second-order backward difference formula is used for the time discretization, and piecewise linear interpolation is used to approximate the interface. We do not assume Lipschitz continuity of the nonlinear coefficient.

Partially penalized immersed finite element methods for parabolic interface problems

We present partially penalized immersed finite element methods for solving parabolic interface problems on Cartesian meshes. Typical semidiscrete and fully discrete schemes are discussed. Error estimates in an energy norm are derived. Numerical examples are provided to support theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1925–1947, 2015

A Matched Alternating Direction Implicit (ADI) Method for Solving the Heat Equation with Interfaces

A novel Douglas alternating direction implicit (ADI) method is proposed in this work to solve a two-dimensional (2D) heat equation with interfaces. The ADI scheme is a powerful finite difference method for solving parabolic equations, due to its unconditional stability and high efficiency. However, it suffers from a serious accuracy reduction in space for interface problems with different materials and nonsmooth solutions. If the jumps in a function and its derivatives are known across the interface, rigorous ADI schemes have been successfully constructed in the literature based on the immersed interface method so that the spatial accuracy can be restored. Nevertheless, the development of accurate and stable ADI methods for general parabolic interface problems with physical interface conditions that describe jumps of a function and its flux, remains unsolved. To overcome this difficulty, a novel tensor product decomposition is proposed in this paper to decouple 2D jump conditions into essentially one-dimensional (1D) ones. These 1D conditions can then be incorporated into the ADI central difference discretization, using the matched interface and boundary technique. Fast algebraic solvers for perturbed tridiagonal systems are developed to maintain the computational efficiency. Stability analysis is conducted through eigenvalue spectrum analysis, which numerically demonstrates the unconditional stability of the proposed ADI method. The matched ADI scheme achieves the first order of accuracy in time and second order of accuracy in space in all tested parabolic interface problems with complex geometries and spatial-temporal dependent jump conditions.

Kinetics Modeling of Parabolic Growth Competing to Linear Declining: A Theoretical Tool to Interpret Corrosion Data in Real World

Corrosion data may be hard to be interpreted in many cases, where the thickening of the oxide scales is simultaneously accompanied by thinning, and the mass gain verse time curves are non-monotone. For example, the thermal growth of the oxide scales (TGO) on metals may compete to basic or acidic fluxing in molten sulphate induced hot corrosion, or to volatilization in water vapor oxidation. We modified the kinetics Tedmon model, which is based on the diffusion controlled parabolic thickening and interface reaction controlled linear thinning of TGO. Both mass gain and scaling rates have been taken into account. It is theoretically deduced that the mass gain of the specimen becomes a maximum (ξ) at a specific time; and then the thickness of TGO (ζ) and the mass gain rate (v0) of the specimen keep constant after a dynamic balance between thickening and thinning of TGO is reached. Formulations have been derived for the determination of the thickening and thinning rate of TGO using measured ξ, ζ and v0.

The 3D Numerical Simulation of Flow Field on Y-Shape Flaring Gate Piers Combined with Stepped Spillway

The energy dissipater of stepped spillway combined with flaring gate pier is widely used in china's hydraulic engineering. The finite volume method is applied to discrete analysis, with the RNG turbulence model and VOF model of water vapor two-phase, iterative solution of geometry reconstruction format unsteady flow to generate free surface. Adopting structured grid for geometric shape, numerically simulated the water vapor two-phase flow from the reservoir to stilling basin. The parabolic water-vapor interface , overall flow pattern, water wings, section depth and other hydraulic characteristics was produced by simulating the three-dimensional flow field.Compared the simulated results of water depth, flow velocity in stilling pool, the board pressure with experiment data, the average error is: the left side depth of 3 # table hole of 7.1%, and the right of 7.4%; the underside flow velocity of 3 # table hole of 5%;1 # table hole stilling pool board pressure of 7.6%,3 # table hole stilling pool board pressure of 6.6%.

A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems

AbstractThis article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time. The method presented here uses immersed finite element (IFE) functions for the discretization in spatial variables that can be carried out over a fixed mesh (such as a Cartesian mesh if desired), and this feature makes it possible to reduce the parabolic equation to a system of ordinary differential equations (ODE) through the usual semi-discretization procedure. Therefore, with a suitable choice of the ODE solver, this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured (Cartesian) mesh. Numerical examples are presented to demonstrate features of this new method.

Selection criterion of stable dendritic growth at arbitrary Péclet numbers with convection

A free dendrite growth under forced fluid flow is analyzed for solidification of a nonisothermal binary system. Using an approach to dendrite growth developed by Bouissou and Pelcé [Phys. Rev. A 40, 6673 (1989)], the analysis is presented for the parabolic dendrite interface with small anisotropy of surface energy growing at arbitrary Péclet numbers. The stable growth mode is obtained from the solvability condition giving the stability criterion for the dendrite tip velocity V and dendrite tip radius ρ as a function of the growth Péclet number, flow Péclet number, and Reynolds number. In limiting cases, the obtained stability criterion presents known criteria for small and high growth Péclet numbers of the solidifying system with and without convective fluid flow.

Semidiscrete Finite Element Methods for Linear and Semilinear Parabolic Problems with Smooth Interfaces: Some New Optimal Error Estimates

Semidiscrete finite element approximations of the linear and semilinear parabolic interface problems are studied in the article. The convergence of the finite element solutions to the exact solutions are analyzed for fitted finite element method with straight interface triangles. Under practical regularity assumptions of the true solution it seems difficult to achieve optimal order convergence with straight interface triangles. In this exposition optimal error estimates in the L 2(L 2) and L 2(H 1) norms are established for linear semidiscrete scheme. Then an extension to the semilinear problem is also considered and related optimal error estimates are achieved. The interfaces are assumed to be smooth for our purpose.

Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients

We consider a model for the propagation of a driven interface through a random field of obstacles. The evolutionequation, commonly referred to as the Quenched Edwards-Wilkinson model, is a semilinear parabolic equationwith a constant driving term and random nonlinearity to model the influence of the obstacle field. For the caseof isolated obstacles centered on lattice points and admitting a random strength with exponential tails, we showthat the interface propagates with a finite velocity for sufficiently large driving force. The proof consistsof a discretization of the evolution equation and a supermartingale estimate akin to the study of branching randomwalks.

Some Error Estimates on the Large Jump Asymptotic Method for Parabolic Iterface Problems

To approximate some problems with strongly discontinuous coefficients, we present the large jump asymptotic method for parabolic interface problems, focus on the derivation and proof of the asymptotic error estimates for our approximation, and show it is of order two. Numerical experiments verify the theoretical results.

Convergence of a finite difference method for solving 2D parabolic interface problems

The convergence of a difference scheme for solving two-dimensional parabolic interface problems with variable coefficients is investigated. An estimate of the rate of convergence in a special discrete W˜22,1 Sobolev norm, compatible with the smoothness of the coefficients and solution, is obtained.