Multipoint Explicit Differencing (MED) for Time Integrations of the Wave Equation

2007 ◽  
Vol 135 (11) ◽  
pp. 3862-3875 ◽  
Author(s):  
Celal S. Konor ◽  
Akio Arakawa
Keyword(s):  
Wave Equation ◽  
Finite Number ◽  
Multilevel Models ◽  
Horizontal Resolution ◽  
Explicit Scheme ◽  
Implicit Scheme ◽  
Courant Number ◽  
Time Step ◽  
Wide Range ◽  
Grid Points

Abstract For time integrations of the wave equation, it is desirable to use a scheme that is stable over a wide range of the Courant number. Implicit schemes are examples of such schemes, but they do that job at the expense of global calculation, which becomes an increasingly serious burden as the horizontal resolution becomes higher while covering a large horizontal domain. If what an implicit scheme does from the point of view of explicit differencing is looked at, it is a multipoint scheme that requires information at all grid points in space. Physically this is an overly demanding requirement because wave propagation in the real atmosphere has a finite speed. The purpose of this study is to seek the feasibility of constructing an explicit scheme that does essentially the same job as an implicit scheme with a finite number of grid points in space. In this paper, a space-centered trapezoidal implicit scheme is used as the target scheme as an example. It is shown that an explicit space-centered scheme with forward time differencing using an infinite number of grid points in space can be made equivalent to the trapezoidal implicit scheme. To avoid global calculation, a truncated version of the scheme is then introduced that only uses a finite number of grid points while maintaining stability. This approach of constructing a stable explicit scheme is called multipoint explicit differencing (MED). It is shown that the coefficients in an MED scheme can be numerically determined by single-time-step integrations of the target scheme. With this procedure, it is rather straightforward to construct an MED scheme for an arbitrarily shaped grid and/or boundaries. In an MED scheme, the number of grid points necessary to maintain stability and, therefore, the CPU time needed for each time step increase as the Courant number increases. Because of this overhead, the MED scheme with a large time step can be more efficient than a usual explicit scheme with a smaller time step only for complex multilevel models with detailed physics. The efficiency of an MED scheme also depends on how the advantage of parallel computing is taken.

scholarly journals A Comparison Study of Explicit and Implicit 3-D Transient Electromagnetic Forward Modeling Schemes on Multi-Resolution Grid

Geosciences ◽  
2021 ◽  
Vol 11 (6) ◽  
pp. 257
Author(s):  
Jingyu Gao ◽  
Maxim Smirnov ◽  
Maria Smirnova ◽  
Gary Egbert
Keyword(s):  
Forward Modeling ◽  
Horizontal Resolution ◽  
Initial Guess ◽  
Explicit Scheme ◽  
Implicit Scheme ◽  
Time Step ◽  
Forward Solution ◽  
Transient Electromagnetic ◽  
Backward Euler ◽  
Background Models

This study compares the efficiency of 3-D transient electromagnetic forward modeling schemes on the multi-resolution grid for various modeling scenarios. We developed time-domain finite-difference modeling based on the explicit scheme earlier. In this work, we additionally implement 3-D transient electromagnetic forward modeling using the backward Euler implicit scheme. The iterative solver is used for solving the system of equations and requires a proper initial guess that has significant effect on the convergence. The standard approach usually employs the solution of a previous time step as an initial guess, which might be too conservative. Instead, we test various initial guesses based on the linear extrapolation or linear combination of the solutions from several previous steps. We build up the implicit scheme forward modeling on the multi-resolution grid, which allows for the adjustment of the horizontal resolution with depth, hence improving the performance of the forward operator. Synthetic examples show the implicit scheme forward modeling using the linearly combined initial guess estimate on the multi-resolution grid additionally reduces the run time compared to the standard initial guess approach. The result of comparison between the implicit scheme developed here with the previously developed explicit scheme shows that the explicit scheme modeling is more efficient for more conductive background models often found in environmental studies. However, the implicit scheme modeling is more suitable for the simulation with highly resistive background models, usually occurring in mineral exploration scenarios. Thus, the inverse problem can be solved using more efficient forward solution depending on the modeling setup and background resistivity.

Stability of wide‐angle absorbing boundary conditions for the wave equation

Geophysics ◽  
1989 ◽  
Vol 54 (9) ◽  
pp. 1153-1163 ◽  
Author(s):  
R. A. Renaut ◽  
J. Petersen
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Least Squares ◽  
Absorbing Boundary Conditions ◽  
Computational Domain ◽  
Wide Angle ◽  
Courant Number ◽  
Absorbing Boundary ◽  
Wide Range ◽  
Artificial Boundaries

Numerical solution of the two‐dimensional wave equation requires mapping from a physical domain without boundaries to a computational domain with artificial boundaries. For realistic solutions, the artificial boundaries should cause waves to pass directly through and thus mimic total absorption of energy. An artificial boundary which propagates waves in one direction only is derived from approximations to the one‐way wave equation and is commonly called an absorbing boundary. Here we investigate order 2 absorbing boundary conditions which include the standard paraxial approximation. Absorption properties are compared analytically and numerically. Our numerical results confirm that the [Formula: see text] or Chebychev‐Padé approximations are best for wide‐angle absorption and that the Chebychev or least‐squares approximations are best for uniform absorption over a wide range of incident angles. Our results also demonstrate, however, that the boundary conditions are stable for varying ranges of Courant number (ratio of time step to grid size). We prove that there is a stability barrier on the Courant number specified by the coefficients of the boundary conditions. Thus, proving stability of the interior scheme is not sufficient. Furthermore, waves may radiate spontaneously from the boundary, causing instability, even if the stability bound on the Courant number is satisfied. Consequently, the Chebychev and least‐squares conditions may be preferred for wide‐angle absorption also.

scholarly journals Development of a semi-Lagrangian advection scheme for the NEMO ocean model (3.1)

2020 ◽  
Vol 13 (9) ◽  
pp. 4379-4398
Author(s):  
Christopher Subich ◽  
Pierre Pellerin ◽  
Gregory Smith ◽  
Frederic Dupont
Keyword(s):  
Ocean Circulation ◽  
Fluid Motion ◽  
Ocean Model ◽  
Global Ocean ◽  
Test Case ◽  
Courant Number ◽  
Time Step ◽  
Fluid Properties ◽  
Grid Points ◽  
Lagrangian Advection

Abstract. As resolutions of ocean circulation models increase, the advective Courant number – the ratio between the distance travelled by a fluid parcel in one time step and the grid size – becomes the most stringent factor limiting model time steps. Some atmospheric models have escaped this limit by using an implicit or semi-implicit semi-Lagrangian formulation of advection, which calculates materially conserved fluid properties along trajectories which follow the fluid motion and end at prescribed grid points. Unfortunately, this formulation is not straightforward in ocean contexts, where the irregular, interior boundaries imposed by the shore and bottom orography are incompatible with traditional trajectory calculations. This work describes the adaptation of the semi-Lagrangian method as an advection module for an operational ocean model. We solve the difficulties of the ocean's internal boundaries by calculating parcel trajectories using a time-exponential formulation, which ensures that all parcel trajectories remain inside the ocean domain despite strong accelerations near the boundary. Additionally, we derive this method in a way that is compatible with the leapfrog time-stepping scheme used in the NEMO-OPA (Nucleus for European Modelling of the Ocean, Océan Parallélisé) ocean model, and we present simulation results for a simplified test case of flow past a model island and for 10-year free runs of the global ocean on the quarter-degree ORCA025 grid.

A New Data-Driven ATHLET – ANSYS CFD Coupling Method for Efficient Simulation of Nuclear Power Plant Circuits

2014 ◽  
Author(s):  
Georgios Theodoridis ◽  
Angel Papukchiev ◽  
Dominik Scholz ◽  
Georg Lerchl
Keyword(s):  
Nuclear Power ◽  
Explicit Scheme ◽  
Implicit Scheme ◽  
Coupling Method ◽  
Data Driven ◽  
Time Step ◽  
Two Phase ◽  
New Approach ◽  
Ansys Cfx ◽  
Coupling Variables

A new data-driven coupling method between the system code ATHLET and ANSYS CFX was developed. The new approach allows for coupled simulations of single and two-phase flow with heat transfer and transport of tracer concentration. Any number of coupling interfaces can be defined with the aid of a graphical user interface. Two coupling schemes have been implemented: an explicit scheme where the coupling variables are exchanged at the end of each CFD time step and a semi-implicit scheme in which the variable exchange is performed for each iteration within each time step. The coupling method was validated with measured data from a single-phase double T-junction mixing experiment. The results of the coupled simulation for the double T-junction case, where 3-D effects are very important, were found in excellent agreement with the experimental data. The semi-implicit scheme was found numerically more accurate and stable than the explicit scheme.

scholarly journals A Unified Convection Scheme (UNICON). Part I: Formulation

2014 ◽  
Vol 71 (11) ◽  
pp. 3902-3930 ◽  
Author(s):  
Sungsu Park
Keyword(s):  
Deep Convection ◽  
Convective Available Potential Energy ◽  
Horizontal Resolution ◽  
Vertical Transport ◽  
Convective Precipitation ◽  
Cold Pool ◽  
Quasi Equilibrium ◽  
Convection Scheme ◽  
Time Step ◽  
Turbulent Eddies

Abstract The author develops a unified convection scheme (UNICON) that parameterizes relative (i.e., with respect to the grid-mean vertical flow) subgrid vertical transport by nonlocal asymmetric turbulent eddies. UNICON is a process-based model of subgrid convective plumes and mesoscale organized flow without relying on any quasi-equilibrium assumptions such as convective available potential energy (CAPE) or convective inhibition (CIN) closures. In combination with a relative subgrid vertical transport scheme by local symmetric turbulent eddies and a grid-scale advection scheme, UNICON simulates vertical transport of water species and conservative scalars without double counting at any horizontal resolution. UNICON simulates all dry–moist, forced–free, and shallow–deep convection within a single framework in a seamless, consistent, and unified way. It diagnoses the vertical profiles of the macrophysics (fractional area, plume radius, and number density) as well as the microphysics (production and evaporation rates of convective precipitation) and the dynamics (mass flux and vertical velocity) of multiple convective updraft and downdraft plumes. UNICON also prognoses subgrid cold pool and mesoscale organized flow within the planetary boundary layer (PBL) that is forced by evaporation of convective precipitation and accompanying convective downdrafts but damped by surface flux and entrainment at the PBL top. The combined subgrid parameterization of diagnostic convective updraft and downdraft plumes, prognostic subgrid mesoscale organized flow, and the feedback among them remedies the weakness of conventional quasi-steady diagnostic plume models—the lack of plume memory across the time step—allowing UNICON to successfully simulate various transitional phenomena associated with convection (e.g., the diurnal cycle of precipitation and the Madden–Julian oscillation).

A method for correcting acoustic finite-difference amplitudes for elastic effects

Geophysics ◽  
2014 ◽  
Vol 79 (4) ◽  
pp. T243-T255 ◽  
Author(s):  
James W. D. Hobro ◽  
Chris H. Chapman ◽  
Johan O. A. Robertsson
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Effective Means ◽  
Cost Effective ◽  
P Wave ◽  
S Waves ◽  
Velocity Models ◽  
Acoustic Simulation ◽  
Wide Range ◽  
Elastic Simulation

We present a new method for correcting the amplitudes of arrivals in an acoustic finite-difference simulation for elastic effects. In this method, we selectively compute an estimate of the error incurred when the acoustic wave equation is used to approximate the behavior of the elastic wave equation. This error estimate is used to generate an effective source field in a second acoustic simulation. The result of this second simulation is then applied as a correction to the original acoustic simulation. The overall cost is approximately twice that of an acoustic simulation but substantially less than the cost of an elastic simulation. Because both simulations are acoustic, no S-waves are generated, so dispersed converted waves are avoided. We tested the characteristics of the method on a simple synthetic model designed to simulate propagation through a strong acoustic impedance contrast representative of sedimentary geology. It corrected amplitudes to high accuracy for reflected arrivals over a wide range of incidence angles. We also evaluated results from simulations on more complex models that demonstrated that the method was applicable in realistic sedimentary models containing a wide range of seismic contrasts. However, its accuracy was reduced for wide-angle reflections from very high impedance contrasts such as a shallow top-salt interface. We examined the influence of modeling at coarse grid resolutions, in which converted S-waves in the equivalent elastic simulation are dispersed. These results provide some validation for the accuracy of the method when applied using finite-difference grids designed for acoustic modeling. The method appears to offer a cost-effective means of modeling elastic amplitudes for P-wave arrivals in a useful range of velocity models. It has several potential applications in imaging and inversion.

scholarly journals A novel semi-implicit scheme for elastodynamics and wave propagation in nearly and truly incompressible solids

2021 ◽  
Author(s):  
Chennakesava Kadapa
Keyword(s):  
Wave Propagation ◽  
Wave Speed ◽  
Critical Time ◽  
Incompressible Material ◽  
Implicit Scheme ◽  
Bulk Wave ◽  
Lumped Mass ◽  
Time Step ◽  
Material Models ◽  
Shear Wave Speed

AbstractThis paper presents a novel semi-implicit scheme for elastodynamics and wave propagation problems in nearly and truly incompressible material models. The proposed methodology is based on the efficient computation of the Schur complement for the mixed displacement-pressure formulation using a lumped mass matrix for the displacement field. By treating the deviatoric stress explicitly and the pressure field implicitly, the critical time step is made to be limited by shear wave speed rather than the bulk wave speed. The convergence of the proposed scheme is demonstrated by computing error norms for the recently proposed LBB-stable BT2/BT1 element. Using the numerical examples modelled with nearly and truly incompressible Neo-Hookean and Ogden material models, it is demonstrated that the proposed semi-implicit scheme yields significant computational benefits over the fully explicit and the fully implicit schemes for finite strain elastodynamics simulations involving incompressible materials. Finally, the applicability of the proposed scheme for wave propagation problems in nearly and truly incompressible material models is illustrated.

Unsteady Conjugate Heat Transfer Investigation of a Multistage Steam Turbine in Warm-Keeping Operation With Hot Air

2018 ◽  
Author(s):  
Piotr Łuczyński ◽  
Dennis Toebben ◽  
Manfred Wirsum ◽  
Wolfgang F. D. Mohr ◽  
Klaus Helbig
Keyword(s):  
Heat Transfer ◽  
Conjugate Heat Transfer ◽  
Flow Patterns ◽  
Operating Conditions ◽  
Angle Of Incidence ◽  
Time Step ◽  
Hot Air ◽  
Wide Range ◽  
Vortex Systems ◽  
Operating Points

In recent decades, the rising share of commonly subsidized renewable energy especially affects the operational strategy of conventional power plants. In pursuit of flexibility improvements, extension of life cycle, in addition to a reduction in start-up time, General Electric has developed a product to warm-keep high/intermediate pressure steam turbines using hot air. In order to optimize the warm-keeping operation and to gain knowledge about the dominant heat transfer phenomena and flow structures, detailed numerical investigations are required. Considering specific warm-keeping operating conditions characterized by high turbulent flows, it is required to conduct calculations based on time-consuming unsteady conjugate heat transfer (CHT) simulations. In order to investigate the warm-keeping process as found in the presented research, single and multistage numerical turbine models were developed. Furthermore, an innovative calculation approach called the Equalized Timescales Method (ET) was applied for the modeling of unsteady conjugate heat transfer (CHT). The unsteady approach improves the accuracy of the stationary simulations and enables the determination of the multistage turbine models. In the course of the research, two particular input variables of the ET approach — speed up factor (SF) and time step (TS) — have been additionally investigated with regard to their high impact on the calculation time and the quality of the results. Using the ET method, the mass flow rate and the rotational speed were varied to generate a database of warm-keeping operating points. The main goal of this work is to provide a comprehensive knowledge of the flow field and heat transfer in a wide range of turbine warm-keeping operations and to characterize the flow patterns observed at these operating points. For varying values of flow coefficient and angle of incidence, the secondary flow phenomena change from well-known vortex systems occurring in design operation (such as passage, horseshoe and corner vortices) to effects typical for windage, like patterns of alternating vortices and strong backflows. Furthermore, the identified flow patterns have been compared to vortex systems described in cited literature and summarized in the so-called blade vortex diagram. The comparison of heat transfer in the form of charts showing the variation of the Nusselt-numbers with respect to changes in angle of incidence and flow coefficients at specific operating points is additionally provided.

On the Friction Coefficient in the Numerical Simulation of Tidal Wave Propagation

2010 ◽  
Author(s):  
Z. Y. Song ◽  
C. Cheng ◽  
F. M. Xu ◽  
J. Kong
Keyword(s):  
Numerical Simulation ◽  
Wave Propagation ◽  
Friction Factor ◽  
Tidal Wave ◽  
Computational Time ◽  
Numerical Dissipation ◽  
Courant Number ◽  
Time Step ◽  
Large Time Step ◽  
The Relationship

Based on the analytical solution of one-dimensional simplified equation of damping tidal wave and Heuristic stability analysis, the precision of numerical solution, computational time and the relationship between the numerical dissipation and the friction dissipation are discussed with different numerical schemes in this paper. The results show that (1) when Courant number is less than unity, the explicit solution of tidal wave propagation has higher precision and requires less computational time than the implicit one; (2) large time step is allowed in the implicit scheme in order to reduce the computational time, but the precision of the solution also reduce and the calculation precision should be guaranteed by reducing the friction factor: (3) the friction factor in the implicit solution is related to Courant number, presented as the determined friction factor is smaller than the natural value when Courant number is larger than unity, and their relationship formula is given from the theoretical analysis and the numerical experiments. These results have important application value for the numerical simulation of the tidal wave.

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