Stable Numerical Solution Research Articles

On uniqueness and stable estimation of multiple parameters in the Cahn–Hilliard equation

We consider the identifiability and stable numerical estimation of multiple parameters in a Cahn–Hilliard model for phase separation. Spatially resolved measurements of the phase fraction are assumed to be accessible, with which the identifiability of single and multiple parameters up to certain scaling invariances is established. A regularized equation error approach is proposed for the stable numerical solution of the parameter identification problems, and convergence of the regularized approximations is proven under reasonable assumptions on the data noise. The viability of the theoretical results and the proposed methods is demonstrated in numerical tests.

A Robust Bubble Growth Solution Scheme for Implementation in CFD Analysis of Multiphase Flows

Although the full form of the Rayleigh–Plesset (RP) equation more accurately depicts the bubble behavior in a cavitating flow than its reduced form, it finds much less application than the latter in the computational fluid dynamic (CFD) simulation due to its high stiffness. The traditional variable time-step scheme for the full form RP equation is difficult to be integrated with the CFD program since it requires a tiny time step at the singularity point for convergence and this step size may be incompatible with time marching of conservation equations. This paper presents two stable and efficient numerical solution schemes based on the finite difference method and Euler method so that the full-form RP equation can be better accepted by the CFD program. By employing a truncation bubble radius to approximate the minimum bubble size in the collapse stage, the proposed schemes solve for the bubble radius and wall velocity in an explicit way. The proposed solution schemes are more robust for a wide range of ambient pressure profiles than the traditional schemes and avoid excessive refinement on the time step at the singularity point. Since the proposed solution scheme can calculate the effects of the second-order term, liquid viscosity, and surface tension on the bubble evolution, it provides a more accurate estimation of the wall velocity for the vaporization or condensation rate, which is widely used in the cavitation model in the CFD simulation. The legitimacy of the solution schemes is manifested by the agreement between the results from these schemes and established ones from the literature. The proposed solution schemes are more robust in face of a wide range of ambient pressure profiles.

Residual error based adaptive method with an optimal variable scaling parameter for RBF interpolation

In infinitely smooth Radial Basis Function (RBF) based interpolation, the scaling parameter plays an important role to obtain an accurate and stable numerical solution. When this method is applied to interpolate a function with sharp gradients, then adaptive methods will also play a significant role in determining an optimal number of centers according to the user desired accuracy. In this article, we test an optimization algorithm developed using the nonlinear optimization to find a scaling parameter for RBF along with an adaptive residual subsampling method [1] RBF interpolation. In this process, at each stage of adoption, the available optimal shape parameters have been obtained by solving the system of non-linear equations.

Mixed Convective Flow in Horizontal Rectangular Enclosure with a Wall Heat Source: A Numerical Study

In this work, we have investigated the heat transfer of a mixed convective flow in horizontal rectangular enclosure having a wall heat source. For the purpose of computing stable and convergent numerical solutions of the flow variables, the governing equations along with the boundary conditions are solved by using the upwind finite volume scheme and a modified SIMPLE algorithm. For discussing the heat transfer from the bottom wall of the enclosure, the average Nusselt numbers are computed from relevant empirical correlations existing in the literature. These correlations are in the range of 1708 ≤ Ra ≤ 108 and 1708 ≤ Ra ≤ 3.5×109 in terms of different Rayleigh numbers for the Newtonian fluids such as air and water. Numerical solutions obtained from this study are compared with available benchmark solutions.

Nonlinear Modal Responses of Damaged Shell Structures: Numerical Prediction and Experimental Validation

The nonlinear finite element (FE) modeling approach has been adopted to model and predict the modal responses of the combined damaged (crack and delamination) layered shell structures. The damaged panel structure has been constructed mathematically using a circular meshing approach of the FE technique to include the crack. Similarly, the sublaminate approach has being used to introduce delamination of the layered structure on a mutual center. The structural distorted geometry and the deformations were modeled through the full geometrical nonlinear strain-displacement (Green–Lagrange) relations in association with higher-order polynomial functions. The modal responses of the damaged structure were obtained through an iterative method in association with the nonlinear FE technique. The predicted response accuracies were established with two-step verifications: that is, the numerical solution stability (elemental sensitivity) and the degree of deviation with published data. The maximum deviation between the developed numerical model and the reference result (first-order shear deformation theory) was 8.3%. The model’s competence and responses were compared with experimental data, with and without damages. Finally, new examples have been solved for different structural geometry-dependent parameters (shell configurations, delamination shapes, crack positions/lengths, end boundaries, etcetera) affecting final modal values. A detailed in-depth understanding of the damage and curvature (unequal/equal curvature) effects on modal responses will be discussed.

An enriched Ciarlet-Raviart scheme for the biharmonic equation

In this paper, we present an enriched Ciarlet-Raviart scheme for the biharmonic equation with variable coefficient on Lipschitz (maybe nonconvex)polyhedral domains. With the enriched finite element space for the Laplacian of the true solution, we manage to prove the discrete $ H^{2} $-stability of numerical solution. Error analysis is provided for solutions with low regularity.

Учет неоднородностей подсеточного масштаба в рамках численной модели глобального распределения электрических полей в ионосфере Земли

Purpose: Creation of a global numerical model of ionospheric electric fields with the possibility of more detailed description of their small-scale features in some limited subareas. Methods: Formulation of a boundary value task with accounting for underlined subdomain for to describe subgrid scale features and the task solution by modern numerical method. Results: Model testing was conducted. In particular, stable numerical solution was obtained for the case of small-scale localized conductivity increase due to additional ionization of the ionosphere by an auroral ray. Practical significance: The model will make it possible to qualify the picture of ionospheric electric field global distribution in those limited subareas which for, there is fuller set of experimental data on the distribution of conductivity and field-aligned currents.

Numerical modeling of wave-porous structure interaction process with an SPH model

<p indent="0mm">This paper presents a smoothed particle hydrodynamics (SPH) model for modeling the fluid-structure interaction of waves and porous structures with high permeability. In the proposed model, the solid and fluid phases are discretized with two types of SPH particles; thus, the model can be applied to study heterogeneous problems directly. We use a varying smoothing length model to preserve the numerical accuracy inside and outside the porous media. The SPH artificial numerical viscosity mode is used in this study to achieve stable numerical solutions. Furthermore, the well-known Forchheimer model has also been used to model the flow in porous media with high Reynolds numbers. With the current model, the experimental observation of the water-level evolutions of a wave-dam interaction benchmark problem has been reproduced. We have also investigated a scenario with a heterogeneous distribution of porosities and compared the simulation results to those of the benchmark problem. Consequently, our model can determine how porous structures can reduce the kinetic energy induced by the waves around the coast.

Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems

Abstract Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability and convergence of time discretizations of nonlinear parabolic equations on an evolving surface, with linearly implicit backward differentiation formulae characteristic trajectories of the surface, for general locally Lipschitz nonlinearities. The discrete maximal $L^p$-regularity is used to prove the boundedness and stability of numerical solutions in the $L^\infty (0,T;W^{1,\infty })$ norm, which is used to bound the nonlinear terms in the stability analysis. Optimal-order error estimates of time discretizations in the $L^\infty (0,T;W^{1,\infty })$ norm is obtained by combining the stability analysis with the consistency estimates.

Stable Numerical Identification of Sources in Non-Homogeneous Media

In this work, we present a numerical algorithm to solve the inverse problem of volumetric sources from measurements on the boundary of a non-homogeneous conductive medium, which is made of conductive layers with constant conductivity in each layer. This inverse problem is ill-posed since there is more than one source that can generate the same measurement. Furthermore, the ill-posedness is due to the fact that small variations (or errors) in the measurement (input data) can produce substantial variations in the identified source location. We propose two steps to solve this inverse problem in some classes of sources: we first recover the harmonic part of the volumetric source, and, in a second step, we compute the non-harmonic part of the source. For the reconstruction of the harmonic part of the source, we follow a variational approach based on the reformulation of the inverse problem as a distributed control problem, for which the cost function incorporates a penalized term with the input data on the boundary. This cost function is minimized by a conjugate gradient algorithm in combination with a finite element discretization. We recover the non-harmonic component of the source using a priori information and an iterative algorithm for some particular classes of sources. To validate the numerical methodology, we develop synthetic examples both in circular (simple) and irregular (complex) regions. The numerical results show that the proposed methodology allows to recover the complete source and produce stable and accurate numerical solutions.

Numerical simulations of interfacial and elastic instabilities

<h2>Abstract</h2> The interplay of elasticity and capillarity in free-surface flows of viscoelastic fluids gives rise to intriguing flow instabilities. We present novel numerical simulations and analyse the physics related to the appearance of surface distortions on polymer melt extrudates, often referred to as "sharkskin" instability, and the abrupt increase in the rise velocity of a bubble in a viscoelastic solution. Regarding the extrusion process, the transition from smooth to wavy extrudate with increasing flow rate is attributed to a Hopf bifurcation, followed by a sequence of period-doubling bifurcations, which eventually lead to elastic turbulence under creeping flow. We suggest that the inception of these waves is related with intense stretch of polymer chains and subsequent recoil at the region where the melt detaches from the solid wall of the die. The ‘velocity discontinuity' of the rising bubble is attributed to a hysteresis loop, triggered by the change of shape of the rear pole into a tip that favors the formation of intense stress gradients, which reduce the fluid viscosity and facilitate the bubble translation. Finally, we discuss the numerical methodology that has allowed us to obtain stable numerical solutions in highly deformed meshes and for high values of the Weissenberg number.

A moving finite element method for solving two-dimensional coupled Burgers’ equations at high Reynolds numbers

The coupled Burgers’ equations at high Reynolds numbers usually have sharp gradients or are discontinuous in the solution. Therefore, it is difficult to obtain analytical solutions. This paper aims to use the moving finite element method proposed by Li et al. (2001) to get stable and high-precision numerical solutions for the coupled Burgers’ equations at high Reynolds numbers. The method decouples the mesh equation and partial differential equation (PDE) into two unrelated parts, mesh reconstruction and PDE solver. The mesh reconstruction constructs the harmonic mapping between the physical and logical domains through an iterative method so that the mesh structure maintains harmonics after multiple numerical integrations. We compute three classic numerical examples. Numerical results show that the moving finite element method effectively solve the coupled Burgers’ equations at high Reynolds numbers, obtain stable numerical results, and achieve higher numerical accuracy. During the evolution of the solution, the mesh is always concentrated in the position where the solution has sharp gradients.

Observational properties of puffy discs: radiative GRMHD spectra of mildly sub-Eddington accretion

ABSTRACT Numerical general relativistic radiative magnetohydrodynamic simulations of accretion discs around a stellar-mass black hole with a luminosity above 0.5 of the Eddington value reveal their stratified, elevated vertical structure. We refer to these thermally stable numerical solutions as puffy discs. Above a dense and geometrically thin core of dimensionless thickness h/r ∼ 0.1, crudely resembling a classic thin accretion disc, a puffed-up, geometrically thick layer of lower density is formed. This puffy layer corresponds to h/r ∼ 1.0, with a very limited dependence of the dimensionless thickness on the mass accretion rate. We discuss the observational properties of puffy discs, particularly the geometrical obscuration of the inner disc by the elevated puffy region at higher observing inclinations, and collimation of the radiation along the accretion disc spin axis, which may explain the apparent super-Eddington luminosity of some X-ray objects. We also present synthetic spectra of puffy discs, and show that they are qualitatively similar to those of a Comptonized thin disc. We demonstrate that the existing xspec spectral fitting models provide good fits to synthetic observations of puffy discs, but cannot correctly recover the input black hole spin. The puffy region remains optically thick to scattering; in its spectral properties, the puffy disc roughly resembles that of a warm corona sandwiching the disc core. We suggest that puffy discs may correspond to X-ray binary systems of luminosities above 0.3 of the Eddington luminosity in the intermediate spectral states.

Novel variable reconstruction and friction term discretisation schemes for hydrodynamic modelling of overland flow and surface water flooding

Due to the recent advances in computing and data acquisition technologies, numerical models solving the 2D shallow water equations (SWEs) are now widely used in predicting overland flow and surface water flooding process in natural catchments. In catchment-scale flood modelling, in addition to the calculation of convective terms that are usually achieved through solving local Riemann problems to capture transient flow dynamics, correct discretisation of source terms is also essential to handle complex domain topography and ensure accurate and stable numerical solutions. The external forces induced by gravity and bed friction create the key source terms that drive the change of momentum in the SWEs and affect the stability and accuracy of the adopted numerical scheme, especially for applications involving very shallow water depth (e.g. wet and dry fronts, overland flows). Herein, new schemes are introduced to reconstruct the face values of flow variables and a fully implicit algorithm is improved to discretise the stiff friction terms, in the context of developing a Godunov-type finite volume hydrodynamic model for accurate and stable simulation of overland flow and surface water flooding involving very shallow water depth. Five analytical test cases are considered to validate the resulting models and the numerical results are compared with those produed by two alternative numerical schemes. The performance of the models are further demonstrated by reproducing a flood event in the 2500 km2 Eden Catchment, UK.

Solving Fredholm integral equation of the first kind using Gaussian process regression

Fredholm integral equation of the first kind is a typical ill-posed problem, and it is usually difficult to obtain a stable numerical solution. In this paper, a new method is proposed to solve Fredholm integral equation using Gaussian process regression (GPR). The key to this method is that the right-hand term of the original integral equation is reconstructed by the GPR model to obtain a new integral equation in a reproducing kernel Hilbert spaces (RKHS). We present an analytical approximate solution of the new equation and prove that it converges to the exact minimal-norm solution of the original equation under the L2-norm. Especially, for the degenerate kernel equation, we obtain an explicit formula of the exact minimal-norm solution. Finally, the proposed method is verified to be very effective in solution accuracy by multiple examples.

Equilibrium stability analysis of hyperbolic shallow water moment equations

In this paper, we analyze the stability of equilibrium manifolds of hyperbolic shallow water moment equations. Shallow water moment equations describe shallow flows for complex velocity profiles which vary in vertical direction and the models can be seen as extensions of the standard shallow water equations. Equilibrium stability is an important property of balance laws that determines the linear stability of solutions in the vicinity of equilibrium manifolds, and it is seen as a necessary condition for stable numerical solutions. After an analysis of the hyperbolic structure of the models, we identify three different stability manifolds based on three different limits of the right‐hand side friction term, which physically correspond to water‐at‐rest, constant‐velocity, and bottom‐at‐rest velocity profiles. The stability analysis then shows that the structural stability conditions are fulfilled for the water‐at‐rest equilibrium and the constant‐velocity equilibrium. However, the bottom‐at‐rest equilibrium can lead to instable modes depending on the velocity profile. Relaxation toward the respective equilibrium manifolds is investigated numerically for different models.

3-D Finite Volume Modeling for LWD Azimuthal Propagation Resistivity Tool With Multiple Annular Antenna Recesses Using Coupled Potentials on Cylindrical Grids

The sensors of logging-while-drilling (LWD) azimuth propagation resistivity (APR) tool are usually installed in the annular antenna recesses filled with ferrite or other magnetic materials on the grooved metal mandrel. In this article, we will improve the 3-D finite volume method (3-D FVM) on cylindrical grids to solve the Helmholtz equations about coupled potentials for precise simulation of the tool responses with multiple antenna recesses. The irregular and novel Yee’s staggered grids are designed to conform to the grooved mandrel surface. The electrical continuation technique is adopted to approximate the perfectly electrical conductor (PEC) boundary condition on the mandrel surface for efficient enhancement of the discretization precision of the potentials around the mandrel. The equivalent conductivity and permeability in heterogeneous elements are determined by the volume average. After that, the parallel direct sparse solver (PARDISO) is used to solve the discretized equation about coupled potentials for effectively ensuring the accuracy and stability of numerical solutions. Finally, we validate the 3-D FVM against results by the hybrid algorithm in homogeneous formation. Through numerical results, we further investigate the influence of both the mandrel and antenna recesses on the LWD tool responses, and its response characteristics in several different environments.

3D Unstructured Spectral Element Method for Frequency-Domain Airborne EM Forward Modeling Based on Coulomb Gauge

In this article, to solve the frequency-domain airborne electromagnetic (EM) modeling problem, we express the electric and magnetic field by the vector magnetic potential and the scalar electric potential and derive the governing equation using the Galerkin weighted residual method under the Coulomb gauge. To avoid the strong singularity of the solution near the transmitting source, we directly solve the relatively slow-changing secondary potential. By introducing the spectral element method (SEM) based on unstructured tetrahedral grids, in which the EM field in each element is characterized by high-order Proriol–Koornwinder–Dubiner (PKD) orthogonal polynomials, we can obtain stable and accurate numerical solutions. By establishing the mapping relationship between the physical domain, the right-angled tetrahedral reference domain, and the orthogonal hexahedral domain, we can easily accomplish the element matrix analysis and calculation. The numerical examples show that, when the SEM is used to simulate the airborne EM response, a high accuracy can be obtained even if a very coarse grid is used. Furthermore, since the tetrahedral grids can easily model the complex boundaries, the SEM based on unstructured grids has significant advantages for simulating complex underground structures.

The Dilemma of Resolving the Low-Frequency Breakdown Problem in Microwave Components via Traditional and Improved Finite-Element Time-Domain Techniques

This paper developed an improved numerical methods for calculating electromagnetic field at low frequencies, and compared the performance with the traditional method. Traditional finite-element time-domain (FETD) is known to suffer from low frequency breakdown (LFB) problem, where system matrix becomes ill-conditioned, and leads to instability of small size electrical structures. In this paper, we proposed a mixed FETD (mFETD) method in resolving the LFB anomaly in the traditional FEM, with a particular reference to transmission line, inductor, and coaxial cable. We considered wave equation of E-field with incorporation of divergence constraint equation (DCE) as a function of Lagrange multiplier using current continuity equation (CCE) and Gauss’s law. For spatial discretization, both nodal basis functions and Curl conforming vector basis functions were selected, and we employed implicit Newmark beta algorithm (NBA) for integration of time. We describe how the components constructed via DCE in system matrix mitigate the singularity impact of the stiffness matrix, which consequently led to significant improvement of the system matrix. Numerical experiment and results demonstrate how the mFETD obtains stable numerical solution and attains faster rate of convergence while using an iterative solver, as such, improves the computational efficiency. Therefore, the mFETD method handles the transient problems in transmission line, which causes LFB anomaly in the traditional FETD, but not efficient in terms of computational time and iteration at solving the coaxial cable problem.

An inverse volatility problem in a degenerate parabolic equation in a bounded domain

&lt;abstract&gt;&lt;p&gt;In this paper, an inverse problem of determining the space-dependent volatility from the observed market prices of options with different strikes is studied. Being different from other inverse volatility problem with classical parabolic equations, we apply the linearization method and introduce some variable substitutions to convert the original problem into an inverse source problem in a degenerate parabolic equation in a bounded area, from which an unknown volatility can be recovered and deficiencies caused by artificial truncation can be solved. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions are deduced, the uniqueness and stability of the minimizer are proved. Then, the Landweber iterative method is used to obtain a stable numerical solution of the inverse problem and some numerical experiments are also performed. The numerical results show that the algorithm which we proposed is robust and the unknown coefficient is recovered quite well.&lt;/p&gt;&lt;/abstract&gt;