scholarly journals Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations

2018 ◽  
Vol 11 (11) ◽  
pp. 4359-4382 ◽  
Author(s):  
Tuomas Kärnä ◽  
Stephan C. Kramer ◽  
Lawrence Mitchell ◽  
David A. Ham ◽  
Matthew D. Piggott ◽  
...  
Keyword(s):  
Discontinuous Galerkin ◽  
Unstructured Grid ◽  
Splitting Method ◽  
Time Integration ◽  
River Estuary ◽  
Coastal Ocean ◽  
Second Order ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Element Discretization

Abstract. Unstructured grid ocean models are advantageous for simulating the coastal ocean and river–estuary–plume systems. However, unstructured grid models tend to be diffusive and/or computationally expensive, which limits their applicability to real-life problems. In this paper, we describe a novel discontinuous Galerkin (DG) finite element discretization for the hydrostatic equations. The formulation is fully conservative and second-order accurate in space and time. Monotonicity of the advection scheme is ensured by using a strong stability-preserving time integration method and slope limiters. Compared to previous DG models, advantages include a more accurate mode splitting method, revised viscosity formulation, and new second-order time integration scheme. We demonstrate that the model is capable of simulating baroclinic flows in the eddying regime with a suite of test cases. Numerical dissipation is well-controlled, being comparable or lower than in existing state-of-the-art structured grid models.

scholarly journals Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations

2018 ◽  
Author(s):  
Tuomas Kärnä ◽  
Stephan C. Kramer ◽  
Lawrence Mitchell ◽  
David A. Ham ◽  
Matthew D. Piggott ◽  
...  
Keyword(s):  
Discontinuous Galerkin ◽  
Unstructured Grid ◽  
Splitting Method ◽  
Time Integration ◽  
River Estuary ◽  
Coastal Ocean ◽  
Second Order ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Element Discretization

Abstract. Unstructured grid ocean models are advantageous for simulating the coastal ocean and river-estuary-plume systems. However, unstructured grid models tend to be diffusive and/or computationally expensive which limits their applicability to real life problems. In this paper, we describe a novel discontinuous Galerkin (DG) finite element discretization for the hydrostatic equations. The formulation is fully conservative and second-order accurate in space and time. Monotonicity of the advection scheme is ensured by using a strong stability preserving time integration method and slope limiters. Compared to previous DG models advantages include a more accurate mode splitting method, revised viscosity formulation, and new second-order time integration scheme. We demonstrate that the model is capable of simulating baroclinic flows in the eddying regime with a suite of test cases. Numerical dissipation is well-controlled, being comparable or lower than in existing state-of-the-art structured grid models.

A New Unconditionally Stable Time Integration Method for Analysis of Nonlinear Structural Dynamics

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
Ali Akbar Gholampour ◽  
Mehdi Ghassemieh ◽  
Mahdi Karimi-Rad
Keyword(s):  
Time Integration ◽  
Second Order ◽  
Numerical Dispersion ◽  
Computational Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Similar Time ◽  
Unconditionally Stable ◽  
Average Acceleration

A new time integration scheme is presented for solving the differential equation of motion with nonlinear stiffness. In this new implicit method, it is assumed that the acceleration varies quadratically within each time step. By increasing the order of acceleration, more terms of the Taylor series are used, which are expected to have responses with better accuracy than the classical methods. By considering this assumption and employing two parameters δ and α, a new family of unconditionally stable schemes is obtained. The order of accuracy, numerical dissipation, and numerical dispersion are used to measure the accuracy of the proposed method. Second order accuracy is achieved for all values of δ and α. The proposed method presents less dissipation at the lower modes in comparison with Newmark's average acceleration, Wilson-θ, and generalized-α methods. Moreover, this second order accurate method can control numerical damping in the higher modes. The numerical dispersion of the proposed method is compared with three unconditionally stable methods, namely, Newmark's average acceleration, Wilson-θ, and generalized-α methods. Furthermore, the overshooting effect of the proposed method is compared with these methods. By evaluating the computational time for analysis with similar time step duration, the proposed method is shown to be faster in comparison with the other methods.

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.

Time Integration of Impact Phenomena in Solid Mechanics: A One-Dimensional Model Problem

1996 ◽  
Author(s):  
V. Chawla ◽  
T. A. Laursen
Keyword(s):  
Finite Element ◽  
Solid Mechanics ◽  
Exact Analytical Solution ◽  
Time Integration ◽  
Dynamic Contact ◽  
Momentum Conservation ◽  
Integration Scheme ◽  
Mass System ◽  
Element Discretization ◽  
Time Integration Scheme

Abstract 1D impact between two identical bars is modeled as a simple spring-mass system as would be generated by a finite element discretization. Some commonly used time integrators are applied to the system to demonstrate defects in the numerical solution as compared to the exact analytical solution. Using energy conservation as the criterion for stability, a new time integration scheme is proposed that imposes a persistency condition for dynamic contact. Finite element simulation with Lagrange multipliers for enforcing the contact constraints shows exact energy and momentum conservation.

scholarly journals Exponential Time Integration and Second-Order Difference Scheme for a Generalized Black-Scholes Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Zhongdi Cen ◽  
Anbo Le ◽  
Aimin Xu
Keyword(s):  
Difference Scheme ◽  
Time Integration ◽  
Second Order ◽  
Integration Scheme ◽  
Uniform Mesh ◽  
Central Difference ◽  
Exponential Time ◽  
Black Scholes ◽  
Time Integration Scheme ◽  
Exponential Time Integration

We apply an exponential time integration scheme combined with a central difference scheme on a piecewise uniform mesh with respect to the spatial variable to evaluate a generalized Black-Scholes equation. We show that the scheme is second-order convergent for both time and spatial variables. It is proved that the scheme is unconditionally stable. Numerical results support the theoretical results.

A Cross Weighted-Residual Time Integration Scheme for Structural Dynamics

2014 ◽  
Vol 14 (06) ◽  
pp. 1450023 ◽  
Author(s):  
Wooram Kim ◽  
Sang-Shin Park ◽  
J. N. Reddy
Keyword(s):  
Structural Dynamics ◽  
Linear Time ◽  
Time Integration ◽  
Velocity Fields ◽  
Inner Product ◽  
Integration Scheme ◽  
Time Interval ◽  
Element Discretization ◽  
Integration Schemes ◽  
Time Integration Scheme

In this article, we develop a novel stable time integration scheme for the transient analysis of structural dynamics problems. A second-order (in time) differential operator equation (e.g. obtained after finite element discretization in space) is written as a pair of first-order equations in terms of displacements and velocities. Then the solution is sought by minimizing the inner product of the residuals in the two equations (an unconventional approach) over typical time interval to obtain a symmetric set of algebraic equations involving displacements and velocities at two subsequent intervals. The new time integration scheme is termed the cross weighted-residual (CWR) time integration scheme because each of the two residuals takes the other one as a weight function in the minimization. The CWR time integration scheme is developed by using a uniform linear time approximation of the displacement and velocity fields to yield only a single step time integration scheme, which is comparable to the Newmark family of time integration scheme. A reduced integration technique is used to prevent velocity locking, which is caused by linear approximation of both the displacement and velocity fields. For the verification of the consistency and the stability, the CWR time integration scheme is tested with single-degree as well as multi-degree of freedom problems. The scheme performs extremely well compared with those of the well-known Newmark family of time integration schemes.

An Explicit Time Integration Scheme Based on B-Spline Interpolation and Its Application in Wave Propagation Analysis

2017 ◽  
Vol 09 (08) ◽  
pp. 1750115 ◽  
Author(s):  
W. B. Wen ◽  
S. Y. Duan ◽  
Y. Tao ◽  
Jun Liang ◽  
Daining Fang
Keyword(s):  
Wave Propagation ◽  
Hyperbolic Equations ◽  
Spline Interpolation ◽  
Time Integration ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Weighted Residual Method ◽  
Explicit Time Integration ◽  
B Spline ◽  
Time Integration Scheme

An explicit time integration scheme for hyperbolic equations is proposed using B-spline interpolation and weighted residual method. It has simple formulation and calculation procedure. With one adjustable algorithmic parameter, new scheme has higher accuracy when compared with other excellent explicit schemes. New scheme has controllable and also desirable period elongation which is verified by theoretical analysis and numerical simulations. Especially, a demonstrative dispersion analysis coupled with the corresponding wave propagation demonstrate the desirable numerical dissipation property and the effectiveness of the proposed scheme for wave propagation problems.

scholarly journals A Time Integration Method Based on Galerkin Weak Form for Nonlinear Structural Dynamics

2019 ◽  
Vol 9 (15) ◽  
pp. 3076
Author(s):  
Qinyan Xing ◽  
Qinghao Yang ◽  
Weixuan Wang
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Weak Form ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Structural Dynamic ◽  
Time Integration Method ◽  
Nonlinear Structural ◽  
Nonlinear Dynamic Equations ◽  
Step Algorithm

This paper presents a step-by-step time integration method for transient solutions of nonlinear structural dynamic problems. Taking the second-order nonlinear dynamic equations as the model problem, this self-starting one-step algorithm is constructed using the Galerkin finite element method (FEM) and Newton–Raphson iteration, in which it is recommended to adopt time elements of degree m = 1,2,3. Based on the mathematical and numerical analysis, it is found that the method can gain a convergence order of 2m for both displacement and velocity results when an ordinary Gauss integral is implemented. Meanwhile, with reduced Gauss integration, the method achieves unconditional stability. Furthermore, a feasible integration scheme with controllable numerical damping has been established by modifying the test function and introducing a special integral rule. Representative numerical examples show that the proposed method performs well in stability with controllable numerical dissipation, and its computational efficiency is superior as well.

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