A multi-phase Mullins–Sekerka system: matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem

1998 ◽  
Vol 128 (3) ◽  
pp. 481-506 ◽  
Author(s):  
Lia Bronsard ◽  
Harald Garcke ◽  
Barbara Stoth
Keyword(s):  
Asymptotic Expansions ◽  
Approximate Solutions ◽  
Implicit Time ◽  
Triple Junctions ◽  
Sharp Interface Model ◽  
Equilibrium Equations ◽  
Time Step ◽  
Time Discretisation ◽  
Geometric Evolution ◽  
Multi Phase

We propose a generalisation of the Mullins–Sekerka problem to model phase separation in multi-component systems. The model includes equilibrium equations in bulk, the Gibbs–Thomson relation on the interfaces, Young's law at triple junctions, together with a dynamic law of Stefan type. Using formal asymptotic expansions, we establish the relationship to a transition layer model known as the Cahn-Hilliard system. We introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds, together with measure theoretical tools. Through an implicit time discretisation, we construct approximate solutions by stepwise minimisation. Under the assumption that there is no loss of area as the time step tends to zero, we show the existence of a weak solution.

GLOBAL EXISTENCE FOR PHASE TRANSITION PROBLEMS VIA A VARIATIONAL SCHEME

2004 ◽  
Vol 01 (04) ◽  
pp. 747-768
Author(s):  
CHRISTIAN ROHDE ◽  
MAI DUC THANH
Keyword(s):  
Phase Transition ◽  
Surface Tension ◽  
Initial Data ◽  
Time Discretization ◽  
Approximate Solutions ◽  
Implicit Time ◽  
Time Step ◽  
Dynamical Phase Transition ◽  
Transition Dynamics ◽  
Variational Scheme

We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this result still holds for non-periodic initial data. Second, we consider a model of phase transition dynamics with only Lipschitz continuous stress–strain function which contains a non-local convolution term to take account of surface tension. We also establish the existence of weak solutions. In both cases the proof relies on implicit time discretization and the analysis of a minimization problem at each time step.

scholarly journals Investigation of Air Pocket Behavior in Pipelines Using Rigid Column Model and Contributions of Time Integration Schemes

Water ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 785
Author(s):  
Arman Rokhzadi ◽  
Musandji Fuamba
Keyword(s):  
Transient Flow ◽  
Time Integration ◽  
Driving Pressure ◽  
Implicit Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Column Model ◽  
Integration Schemes ◽  
Backward Euler

This paper studies the air pressurization problem caused by a partially pressurized transient flow in a reservoir-pipe system. The purpose of this study is to analyze the performance of the rigid column model in predicting the attenuation of the air pressure distribution. In this regard, an analytic formula for the amplitude and frequency will be derived, in which the influential parameters, particularly, the driving pressure and the air and water lengths, on the damping can be seen. The direct effect of the driving pressure and inverse effect of the product of the air and water lengths on the damping will be numerically examined. In addition, these numerical observations will be examined by solving different test cases and by comparing to available experimental data to show that the rigid column model is able to predict the damping. However, due to simplified assumptions associated with the rigid column model, the energy dissipation, as well as the damping, is underestimated. In this regard, using the backward Euler implicit time integration scheme, instead of the classical fourth order explicit Runge–Kutta scheme, will be proposed so that the numerical dissipation of the backward Euler implicit scheme represents the physical dissipation. In addition, a formula will be derived to calculate the appropriate time step size, by which the dissipation of the heat transfer can be compensated.

scholarly journals A Semi-Implicit Version of the MPAS-Atmosphere Dynamical Core

2015 ◽  
Vol 143 (9) ◽  
pp. 3838-3855 ◽  
Author(s):  
Steven Sandbach ◽  
John Thuburn ◽  
Danail Vassilev ◽  
Michael G. Duda
Keyword(s):  
Time Integration ◽  
Standard Test ◽  
Newton Iteration ◽  
Atmospheric Modeling ◽  
Implicit Time ◽  
Integration Scheme ◽  
Baroclinic Wave ◽  
Time Step ◽  
Integration Schemes ◽  
Helmholtz Problem

Abstract An important question for atmospheric modeling is the viability of semi-implicit time integration schemes on massively parallel computing architectures. Semi-implicit schemes can provide increased stability and accuracy. However, they require the solution of an elliptic problem at each time step, creating concerns about their parallel efficiency and scalability. Here, a semi-implicit (SI) version of the Model for Prediction Across Scales (MPAS) is developed and compared with the original model version, which uses a split Runge–Kutta (SRK3) time integration scheme. The SI scheme is based on a quasi-Newton iteration toward a Crank–Nicolson scheme. Each Newton iteration requires the solution of a Helmholtz problem; here, the Helmholtz problem is derived, and its solution using a geometric multigrid method is described. On two standard test cases, a midlatitude baroclinic wave and a small-planet nonhydrostatic gravity wave, the SI and SRK3 versions produce almost identical results. On the baroclinic wave test, the SI version can use somewhat larger time steps (about 60%) than the SRK3 version before losing stability. The SI version costs 10%–20% more per step than the SRK3 version, and the weak and strong scalability characteristics of the two versions are very similar for the processor configurations the authors have been able to test (up to 1920 processors). Because of the spatial discretization of the pressure gradient in the lowest model layer, the SI version becomes unstable in the presence of realistic orography. Some further work will be needed to demonstrate the viability of the SI scheme in this case.

scholarly journals A New Approach to Approximate Solutions for Nonlinear Differential Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Safia Meftah
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Small Parameter ◽  
Perturbation Method ◽  
Periodic Solutions ◽  
Asymptotic Expansions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Approximation Methods ◽  
New Approach

The question discussed in this study concerns one of the most helpful approximation methods, namely, the expansion of a solution of a differential equation in a series in powers of a small parameter. We used the Lindstedt-Poincaré perturbation method to construct a solution closer to uniformly valid asymptotic expansions for periodic solutions of second-order nonlinear differential equations.

scholarly journals Efficient Approximate Techniques for Integrating Stochastic Differential Equations

2006 ◽  
Vol 134 (10) ◽  
pp. 3006-3014 ◽  
Author(s):  
James A. Hansen ◽  
Cecile Penland
Keyword(s):  
Time Series ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Deterministic Model ◽  
Probability Distributions ◽  
Approximate Solutions ◽  
Integration Scheme ◽  
Time Step ◽  
Integration Schemes ◽  
Runge Kutta

Abstract The delicate (and computationally expensive) nature of stochastic numerical modeling naturally leads one to look for efficient and/or convenient methods for integrating stochastic differential equations. Concomitantly, one may wish to sensibly add stochastic terms to an existing deterministic model without having to rewrite that model. In this note, two possibilities in the context of the fourth-order Runge–Kutta (RK4) integration scheme are examined. The first approach entails a hybrid of deterministic and stochastic integration schemes. In these examples, the hybrid RK4 generates time series with the correct climatological probability distributions. However, it is doubtful that the resulting time series are approximate solutions to the stochastic equations at every time step. The second approach uses the standard RK4 integration method modified by appropriately scaling stochastic terms. This is shown to be a special case of the general stochastic Runge–Kutta schemes considered by Ruemelin and has global convergence of order one. Thus, it gives excellent results for cases in which real noise with small but finite correlation time is approximated as white. This restriction on the type of problems to which the stochastic RK4 can be applied is strongly compensated by its computational efficiency.

A FULLY IMPLICIT SOLVER OF 3-D EULER EQUATIONS ON MULTIBLOCK CURVILINEAR GRIDS

2005 ◽  
Vol 19 (28n29) ◽  
pp. 1483-1486 ◽  
Author(s):  
HAI-QING SI ◽  
TONG-GUANG WANG ◽  
XIAO-YUN LUO
Keyword(s):  
Euler Equations ◽  
Three Dimensional ◽  
Implicit Time ◽  
Tvd Scheme ◽  
Time Step ◽  
Vector Operation ◽  
Fully Implicit ◽  
Curvilinear Grids ◽  
The Matrix ◽  
Implicit Solver

A fully implicit unfactored algorithm for three-dimensional Euler equations is developed and tested on multi-block curvilinear meshes. The convective terms are discretized using an upwind TVD scheme. The large sparse linear system generated at each implicit time step is solved by GMRES* method combined with the block incomplete lower-upper preconditioner. In order to reduce the memory requirements and the matrix-vector operation counts, an approximate method is used to derive the Jacobian matrix, which only costs half of the computational efforts of the exact Jacobian calculation. The comparison between the numerical results and the experimental data shows good agreement, which demonstrates that the implicit algorithm presented is effective and efficient.

On the Validity and Sensitivity of CFD Simulations for a Deterministic Breaking Wave Impact on a Semi Submersible

2018 ◽  
Author(s):  
Henry Bandringa ◽  
Joop A. Helder
Keyword(s):  
Free Surface ◽  
Absorbing Boundary Conditions ◽  
Implicit Time ◽  
Breaking Wave ◽  
Computational Domain ◽  
Wave Impact ◽  
Time Step ◽  
Cfd Simulations ◽  
Detailed Model ◽  
Run Up

To assess the integrity and safety of structures offshore, prediction of run-up, green water, and impact loads needs to be made during the structure’s design. For predicting these highly non-linear phenomena, most of the offshore industry relies on detailed model testing. In the last couple of years however, CFD simulations have shown more and more promising results in predicting these events, see for instance [1]–[4]. To obtain confidence in the accuracy of CFD simulations in the challenging field of extreme wave impacts, a proper validation of such CFD tools is essential. In this paper two CFD tools are considered for the simulation of a deterministic breaking wave impact on a fixed semi submersible, resulting in flow phenomena like wave run-up, horizontal wave impact and deck impacts. Hereby, one of the CFD tools applies an unstructured gridding approach and implicit free-surface reconstruction, and uses an implicit time integration with a fixed time step. The other CFD tool explicitly reconstructs the free surface on a structured grid and integrates the free surface explicitly in time, using a variable time step. The presented simulations use a compact computational domain with wave absorbing boundary conditions and local grid refinement to reduce CPU time. Besides a typical verification and validation of the results, for one of the CFD tools a sensitivity study is performed in which the influence of small variations in the incoming breaking wave on the overall results is assessed. Such an analysis should provide the industry more insight in the to-be-expected sensitivity (and hence uncertainty) of CFD simulations for these type of applications. Experiments carried out by MARIN are used to validate all the presented simulation results.

Numerical simulation of higher-order diffusion-wave equations of variable coefficients using the meshless spectral method

2020 ◽  
pp. 2150010
Author(s):  
Manzoor Hussain ◽  
Sirajul Haq
Keyword(s):  
Wave Equations ◽  
Interpolation Method ◽  
Variable Coefficients ◽  
Higher Order ◽  
Implicit Time ◽  
Test Problems ◽  
Time Step ◽  
Diffusion Wave ◽  
Time Step Size ◽  
Time Stepping

In this paper, meshless spectral interpolation technique using implicit time stepping scheme is proposed for the numerical simulations of time-fractional higher-order diffusion wave equations (TFHODWEs) of variable coefficients. Meshless shape functions, obtained from radial basis functions (RBFs) and point interpolation method (PIM), are used for spatial approximation. Central differences coupled with quadrature rule of [Formula: see text] are employed for fractional temporal approximation. For advancement of solution, an implicit time stepping scheme is then invoked. Simulations performed for different benchmark test problems feature good agreement with exact solutions. Stability analysis of the proposed method is theoretically discussed and computationally validated to support the analysis. Accuracy and efficiency of the proposed method are assessed via [Formula: see text], [Formula: see text] and [Formula: see text] error norms as well as number of nodes [Formula: see text] and time step-size [Formula: see text].

scholarly journals Hybrid Fixed-Point Fixed-Stress Splitting Method for Linear Poroelasticity

Geosciences ◽  
2019 ◽  
Vol 9 (1) ◽  
pp. 29 ◽  
Author(s):  
Paul M. Delgado ◽  
V. M. Krushnarao Kotteda ◽  
Vinod Kumar
Keyword(s):  
Fixed Point ◽  
Asymptotic Solution ◽  
Relative Error ◽  
Operator Splitting ◽  
Splitting Method ◽  
Approximate Solutions ◽  
Time Step ◽  
Operator Splitting Method ◽  
The Stability ◽  
Stability And Consistency

Efficient and accurate poroelasticity models are critical in modeling geophysical problems such as oil exploration, gas-hydrate detection, and hydrogeology. We propose an efficient operator splitting method for Biot’s model of linear poroelasticity based on fixed-point iteration and constrained stress. In this method, we eliminate the constraint on time step via combining our fixed-point approach with a physics-based restraint between iterations. Three different cases are considered to demonstrate the stability and consistency of the method for constant and variable parameters. The results are validated against the results from the fully coupled approach. In case I, a single iteration is used for continuous coefficients. The relative error decreases with an increase in time. In case II, material coefficients are assumed to be linear. In the single iteration approach, the relative error grows significantly to 40% before rapidly decaying to zero. This is an artifact of the approximate solutions approaching the asymptotic solution. The error in the multiple iterations oscillates within 10 − 6 before decaying to the asymptotic solution. Nine iterations per time step are enough to achieve the relative error close to 10 − 7 . In the last case, the hybrid method with multiple iterations requires approximately 16 iterations to make the relative error 5 × 10 − 6 .

A finite‐element solution for the transient electromagnetic response of an arbitrary two‐dimensional resistivity distribution

Geophysics ◽  
1986 ◽  
Vol 51 (7) ◽  
pp. 1450-1461 ◽  
Author(s):  
Y. Goldman ◽  
C. Hubans ◽  
S. Nicoletis ◽  
S. Spitz
Keyword(s):  
Finite Element ◽  
Sparse Matrix ◽  
Equation System ◽  
Electromagnetic Response ◽  
Implicit Time ◽  
Two Dimensional ◽  
Time Step ◽  
Transient Electromagnetic ◽  
Exact Boundary ◽  
The Time Domain

We present a numerical method for solving Maxwell’s equations in the case of an arbitrary two‐dimensional resistivity distribution excited by an infinite current line. The electric field is computed directly in the time domain. The computations are carried out in the lower half‐space only because exact boundary conditions are used on the free surface. The algorithm follows the finite‐element approach, which leads (after space discretization) to an equation system with a sparse matrix. Time stepping is done with an implicit time scheme. At each time step, the solution of the equation system is provided by the fast system ICCG(0). The resulting algorithm produces good results even when large resistivity contrasts are involved. We present a test of the algorithm’s performance in the case of a homogeneous earth. With a reasonable grid, the relative error with respect to the analytical solution does not exceed 1 percent, even 2 s after the source is turned off.

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