Implicit Time Discretization Scheme Research Articles

Optimally Convergent HDG Method for Third-Order Korteweg–de Vries Type Equations

We develop and analyze a new hybridizable discontinuous Galerkin (HDG) method for solving third-order Korteweg-de Vries type equations. The approximate solutions are defined by a discrete version of a characterization of the exact solution in terms of the solutions to local problems on each element which are patched together through transmission conditions on element interfaces. We prove that the semi-discrete scheme is stable with proper choices of stabilization function in the numerical traces. For the linearized equation, we carry out error analysis and show that the approximations to the exact solution and its derivatives have optimal convergence rates. In numerical experiments, we use an implicit scheme for time discretization and the Newton-Raphson method for solving systems of nonlinear equations, and observe optimal convergence rates for both the linear and the nonlinear third-order equations.

Simulation of coupled structural-acoustic response with dynamic damage evolution

A novel time-domain method for simulating dynamic damage evolution in a coupled structural-acoustic system is presented. The system is derived via the theory of continuum damage mechanics, and incorporates standard damage evolution models, but is readily extendible to more exotic formulations. The overall solution method is staggered, solving for the dynamic damage evolution first with an explicit step, and then using the new values in the coupled computation of the structural-acoustic system. The spatial domain is discretized using a mixed finite element method, and the temporal domain is discretized with a higher-order implicit time discretization scheme. Efforts toward fully coupled verification of the solution algorithm are presented, as are validation studies for cases without evolving damage. Applications with evolving damage are presented, and present a first principles study of changes in the structural acoustic response to dynamically evolving damage in the structure. Special attention is given to brittle fracture. Examples of downstream usage of the evolving structural response are discussed in the concluding remarks.

Symmetrization for fractional Neumann problems

In this paper we complement the program concerning the application of symmetrization methods to nonlocal PDEs by providing new estimates, in the sense of mass concentration comparison, for solutions to linear fractional elliptic and parabolic PDEs with Neumann boundary conditions. These results are achieved by employing suitable symmetrization arguments to the Stinga–Torrea local extension problems, corresponding to the fractional boundary value problems considered. Sharp estimates are obtained first for elliptic equations and a certain number of consequences in terms of regularity estimates is then established. Finally, a parabolic symmetrization result is covered as an application of the elliptic concentration estimates in the implicit time discretization scheme.

The two-obstacle problem for the parabolic biharmonic equation

We consider a two-obstacle problem for the parabolic biharmonic equation in a bounded domain. We prove long time existence of solutions via an implicit time discretization scheme, and we investigate the regularity properties of solutions.

Aeroelastic Analysis of a Typical Section using Euler and Navier-Stokes Mesh-less Method

The main aim of this paper is to develop an efficient aeroelastic tool for predicting the flutter speed of a typical section in transonic regime. An implicit mesh-less method based on the Euler and Navier-Stokes equations are conducted to simulate the transonic fluid flow around an airfoil. This technique is applied directly to the differential form of the aerodynamic governing equations and the time integration is carried out using a dual-time implicit time discretization scheme. The capabilities of the flow solution method are demonstrated by flow computations around NACA0012 airfoil at different flow conditions. For structural dynamics simulation, a typical section model with pitching and plunging motion capability is considered. Finally, the aeroelastic analysis of the 2D model is performed by the consecutive simulation of both structural and aerodynamic domains. Also, the effect of viscosity and time interval choice between two structural and aerodynamic solvers on the flutter instability is studied. The comparison between the obtained results and those available in the literature shows the good accuracy of the present method.

Influence of second-phase inclusions on the electro-deformation of ferroelectric ceramics

Theoretical estimates are given for the overall dissipative response of ferroelectric ceramics with second-phase inclusions, under arbitrary electromechanical loading histories. The ferroelectric behavior of the constituent phases is described via a stored energy density and a dissipation potential in accordance with the theory of generalized standard materials. An implicit time-discretization scheme is used to generate a variational representation of the overall response in terms of a single incremental potential. Estimates are then generated by constructing sequentially laminated microgeometries of particulate type whose overall incremental potential can be computed exactly. Because they are realizable, by construction, these estimates are guaranteed to conform with any material constraints, to satisfy all pertinent bounds, and to exhibit the required convexity properties with no duality gap. By way of example, the theory is used to study the influence of metallic particles and of microcavities on the electro-deformability of a lead zirconate titanate. In particular, the role of remanent polarization fluctuations on the piezoelectric properties is assessed.

Simulation of coupled structural-acoustic response with dynamic damage evolution

A novel time-domain method for simulating dynamic damage evolution in a coupled structural-acoustic system is presented. The system is derived via the theory of continuum damage mechanics, and incorporates standard damage evolution models. The overall solution method is staggered, solving for the dynamic damage evolution first with an explicit step, and then using the new values in the coupled computation of the structural-acoustic system. The spatial domain is discretized using a mixed finite element method, and the temporal space is discretized with a higher-order implicit time discretization scheme. Efforts toward fully coupled verification of the solution algorithm are presented, as are validation studies for cases without evolving damage. Applications with evolving damage are presented, and present a first principles study of changes in the structural acoustic response to dynamically evolving damage in the structure. Examples of downstream usage of the evolving structural response are discussed in the concluding remarks.

Horizontally Explicit and Vertically Implicit (HEVI) Time Discretization Scheme for a Discontinuous Galerkin Nonhydrostatic Model

Abstract A two-dimensional nonhydrostatic (NH) atmospheric model based on the compressible Euler system has been developed in the (x, z) Cartesian domain. The spatial discretization is based on a nodal discontinuous Galerkin (DG) method with exact integration. The orography is handled by the terrain-following height-based coordinate system. The time integration uses the horizontally explicit and vertically implicit (HEVI) time-splitting scheme, which is introduced to address the stringent restriction on the explicit time step size due to a high aspect ratio between the horizontal (x) and vertical (z) spatial discretization. The HEVI scheme is generally based on the Strang-type operator-split approach, where the horizontally propagating waves in the Euler system are solved explicitly while the vertically propagating waves are treated implicitly. As a consequence, the HEVI scheme relaxes the maximum allowed time step to be mainly determined by the horizontal grid spacing. The accuracy of the HEVI scheme is rigorously compared against that of the explicit strong stability-preserving (SSP) Runge–Kutta (RK) scheme using several NH benchmark test cases. The HEVI scheme shows a second-order temporal convergence, as expected. The results of the HEVI scheme are qualitatively comparable to those of the SSP-RK3 scheme. Moreover, the HEVI DG formulation can also be seamlessly extended to account for the second-order diffusion as in the case of the standard SSP-RK DG formulation. In the presence of orography, the HEVI scheme produces high quality results, which are visually identical to those produced by the SSP-RK3 scheme.

ADI Schemes for Pricing American Options under the Heston Model

In this article, a simple, effective adaptation of Alternating Direction Implicit time discretization schemes is proposed for the numerical pricing of American-style options under the Heston model via a partial differential complementarity problem. The stability and convergence of the new methods are extensively investigated in actual, challenging applications. In addition, a relevant theoretical result is proved.

Incremental variational approach for time dependent deformation in clayey rock

An extension of Lahellec (2007) incremental variational approach to geomaterials is proposed. By using an implicit time-discretization scheme, the evolution equations describing the small strains constitutive behavior of the phases can be reduced to the minimization of an incremental energy function. This minimization problem is rigorously equivalent to a nonlinear thermoelastic problem with a transformation strain that is a heterogeneous field. The new approach proposed is for Callovo-Oxfordian argillites and considers the specific properties of geomaterials such as compressibility-dilatancy transition and influence of confining pressure. An isotropic, kinematic hardening effect is also considered for the more general cases. The accuracy of the model is assessed by comparison with FE and experimental results.

Modeling two-phase ferroelectric composites by sequential laminates

Theoretical estimates are given for the overall dissipative response of two-phase ferroelectric composites with complex particulate microstructures under arbitrary loading histories. The ferroelectric behavior of the constituent phases is described via a stored energy density and a dissipation potential in accordance with the theory of generalized standard materials. An implicit time-discretization scheme is used to generate a variational representation of the overall response in terms of a single incremental potential. Estimates are then generated by constructing sequentially laminated microgeometries of particulate type whose overall incremental potential can be computed exactly. Because they are realizable, by construction, these estimates are guaranteed to conform with any material constraints, to satisfy all pertinent bounds and to exhibit the required convexity properties with no duality gap. Predictions for representative composite and porous systems are reported and discussed in the light of existing experimental data.

Approximation results and subspace correction algorithms for implicit variational inequalities

This paper deals with the mathematical analysis and the subspace approximation of a system of variational inequalities representing a unified approach to several quasistatic contact problems in elasticity. Using an implicit time discretization scheme and some estimates, convergence properties of the incremental solutions and existence results are presented for a class of abstract implicit evolution variational inequalities involving a nonlinear operator. To solve the corresponding semi-discrete and the fully discrete problems, some general subspace correction algorithms are proposed, for which global convergence is analyzed and error estimates are established.

Finite element approximation of nonlinear transient magnetic problems involving periodic potential drop excitations

This paper deals with the computation of nonlinear 2D transient magnetic fields when the data concerning the electric current sources involve potential drop excitations. In the first part, a mathematical model is stated, which is solved by an implicit time discretization scheme combined with a finite element method for space approximation. The second part focuses on developing a numerical method to compute periodic solutions by determining a suitable initial current which avoids large simulations to reach the steady state. This numerical method leads to solve a nonlinear system of equations which requires to approximate several nonlinear and linear magnetostatic problems. The proposed methods are first validated with an axisymmetric example and sinusoidal source having an analytical solution. Then, we show the saving in computational effort that this methodology offers to approximate practical problems specially with pulse-width modulation (PWM) voltage supply.

An iterative finite element time-domain method for simulating three-dimensional electromagnetic diffusion in earth

Geophysical Journal International An iterative finite element time-domain method for simulating three-dimensional electromagnetic diffusion in earth Evan Schankee Um, 1 Jerry M. Harris 2 and David L. Alumbaugh 3 1 Earth Sciences Division, Geophysics Department, Lawrence Berkeley National Laboratory, Berkeley, CA, USA. E-mail: evanum@gmail.com USA 3 Chevron Energy Technology, 6001 Bollinger Canyon Road, San Ramon, CA 94583, USA 2 Geophysics Department, Stanford University, Stanford, CA, SUMMARY An iterative finite element time-domain (FETD) method has been developed for simulating transient electromagnetic fields in 3-D diffusive earth media and has been verified through comparisons with analytic and finite-difference time-domain solutions. The adaptive time step doubling (ATSD) method plays an important role in reducing solution run-time by allowing large time steps in late time when high-frequency electric fields are increasingly attenuated in the Earth. We demonstrate that for the ATSD method to work effectively, the conductivity of the air and a drop tolerance of a preconditioner should be carefully selected. The conductivity of the air should not be too large for accurate simulations but also not too small for avoid- ing ill conditioning that results in error amplification in the ATSD method. A proper drop tolerance keeps the eigenvalues of the preconditioned FETD matrices clustered when a time step size is successively doubled during the ATSD processes, resulting in the convergence of iterative solutions with the reasonable number of iterations. A rule of thumb for determining the conductivity of the air and the drop tolerance has been presented. We also present the simultaneous multiple-sources modelling (SM 2 ) approach. The SM 2 approach simultaneously advances the electric fields excited by multiple individual sources in a single time stepping loop. This approach allows multiple sources to share the same preconditioner in the time stepping loop and improves the simulation efficiency per a survey line. Key words: Numerical solutions; Electromagnetic theory; Marine electromagnetics. I N T R O D U C T I O N Transient electromagnetic (TEM) methods have been widely used in a wide range of geophysical research such as deep crustal stud- ies (e.g. Ho¨ rdt et al. 1992; Ho¨ rdt et al. 2002), energy explorations (e.g. Strack 1992), environmental investigations (e.g. Auken et al. ´ rnason et al. 2010). In- 2006) and geothermal explorations (e.g. A terpretations of TEM data in complex geological environments in- creasingly resort to multidimensional inverse modelling. Because forward modelling is a major numerical bottleneck for the inverse modelling, it is important to develop efficient and accurate forward modelling algorithms. Introduced by Yee (1966) in electrical engineering, finite- difference time-domain (FDTD) methods have become one of the standard tools used to simulate TEM fields in geophysics. Their popularity is mainly based on the fact that FDTD methods are relatively easy to implement, easy to use and can provide rea- sonably accurate solutions over a wide range of geophysics prob- lems. For example, Goldman & Stoyer (1983) present 2-D finite- difference modelling of TEM fields in simplified axially symmetric earth media. Wang & Hohmann (1993) develop a 3-D FDTD al- gorithm that advances TEM fields in time using an explicit time stepping approach. Commer & Newman (2004) introduce the par- allel version of Wang & Hohmann (1993) along with new algo- rithmic features such as parallel upward continuation procedures and parallel time stepping. The parallel version has been em- bedded to a 3-D TEM inversion algorithm (Newman & Commer Despite their popularity, the 3-D FDTD methods also have well known disadvantages. For example, their explicit time stepping ap- proach requires small time step sizes to satisfy stability conditions especially when there are large conductivity contrasts. To overcome this issue, Haber et al. (2004) use an unconditionally stable implicit time discretization scheme in their finite volume approach (we also employ an implicit time discretization scheme in this paper). Be- sides, in FDTD modelling, complex geological structures such as seafloor bathymetry, reservoirs and salt domes need to be approx- imated by small stair steps when the structures do not confirm to rectangular cells. Therefore, this approximation approach can quickly increase FDTD problem sizes; FDTD problems resulting from realistic 3-D earth models are usually solved in massively parallel computing environments (Commer & Newman 2004; Newman & Commer 2005; Commer et al. 2008). Furthermore, such stair steps can introduce discretization errors into numerical modelling results especially when sources and receivers are placed on or very close to the complex surface described by the stair steps.

3D implicit mesh-less method for compressible flow calculations

An implicit mesh-less method is developed for calculation of compressible flows around three dimensional complex geometries. The algorithm is applied directly to the differential form of the governing equations using least-square formulation. A dual-time implicit time discretization scheme is developed, and the computational efficiency is enhanced by adopting accelerating techniques, such as local time stepping, residual smoothing and enthalpy damping. Two different artificial dissipation techniques are employed for stability preservation and it is shown that the scalar one is more efficient in terms of accuracy and computational time. The capabilities of the method are demonstrated by flow computations around different geometries under subsonic and transonic flow conditions. Results are presented which indicate good agreement with experimental and other reliable numerical data. The method is shown to reduce computational time by about 50% compared with the alternative explicit method.

Mean curvature flow with obstacles

We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity of the obstacles, in the two-dimensional case we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss an application of this result to the positive mean curvature flow.

Continuity of solutions of a quasilinear hyperbolic equation with hysteresis

This paper is devoted to the investigation of quasilinear hyperbolic equations of first order with convex and nonconvex hysteresis operator. It is shown that in the nonconvex case the equation, whose nonlinearity is caused by the hysteresis term, has properties analogous to the quasilinear hyperbolic equation of first order. Hysteresis is represented by a functional describing adsorption and desorption on the particles of the substance. An existence result is achieved by using an approximation of implicit time discretization scheme, a priori estimates and passage to the limit; in the convex case it implies the existence of a continuous solution.

The 3-D time-dependent transport code TORT-TD and its coupling with the 3D thermal-hydraulic code ATTICA3D for HTGR applications

The application of the time-dependent 3-D discrete-ordinates neutron transport code TORT-TD to HTGR of pebble bed type and its coupling with the 3-D thermal-hydraulic code ATTICA3D is described. TORT-TD, developed at GRS, solves the time-dependent multi-group SN transport equation in both Cartesian and curvilinear geometry using a fully implicit time discretization scheme. To provide a detailed 3-D core model for pebble bed type HTGRs, TORT-TD has been coupled with the 3-D thermal-hydraulic code ATTICA3D which is developed at IKE, University of Stuttgart. ATTICA3D provides a 3-D modeling of heat conduction and convection using a porous media approach. In this paper, the TORT-TD/ATTICA3D coupling methodology is described. The PBMR-268 and PBMR-400 benchmark problems have been chosen as first test cases for TORT-TD and the coupled code TORT-TD/ATTICA3D. In TORT-TD, the control rods in the side reflector have been modeled both as grey curtain and spatially resolved. Results for first coupled steady-state and transient calculations of the PBMR-400 benchmark are shown.

Water flow through unsaturated porous media with hysteresis

The paper is devoted to the investigation of a parabolic equation with the Preisach operator under the time derivative. The model equation appears in the context of soil water hysteresis. Under suitable assumptions an existence result is obtained by using an implicit time discretization scheme, a priori estimates and passage to the limit in the convexity domain of the Preisach operator.

Computational Study on a Squid-Like Underwater Robot with Two Undulating Side Fins

Abstract The undulating fin propulsion system is an instance of the bio-inspired propulsion systems. In the current study, the swimming motion of a squid-like robot with two undulating side fins, mimicking those of a Stingray or a Cuttlefish, was investigated through flow computation around the body. We used the finite analytic method for space discretization and Euler implicit scheme for time discretization along with the PISO algorithm for velocity pressure coupling. A body-fitted moving grid was generated using the Poisson equation at each time step. Based on the computed results, we discussed the features of the flow field and hydrodynamic forces acting on the body and fin. A simple relationship among the fin's principal dimensions was established. Numerical computation was done for various aspect ratios, fin angles and frequencies in order to validate the proposed relationship among principal dimensions. Subsequently, the relationship was examined base on the distribution of pressure difference between upper and lower surfaces and the distribution of the thrust force. In efficiency calculations, the undulating fins showed promising results. Finally, for the fin, the open characteristics from computed data showed satisfactory conformity with the experimental results.