Adaptive Grid Technique Research Articles

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 15: patch-adaptive simulation of example transient experiments described by Nernst–Planck–electroneutrality equations in one-dimensional space geometry

The patch-adaptive strategy for electrochemical kinetic simulations, introduced in Part 5 of the present series of papers, with extensions described in Part 14, is applied to five typical examples of time-dependent models of transient experiments represented by Nernst–Planck–electroneutrality equation systems in one-dimensional space geometry. The models describe: potential step chronoamperometry for a charge neutralisation reaction at a planar electrode in the absence of the supporting electrolyte; potential step chronoamperometry for a planar electrode coated with a polymer film under conditions of migration–diffusion of redox ions; potential step chronoamperometry of uncharged species at a hemispherical electrode in the presence of a low concentration of supporting electrolyte; cyclic voltammetry for an electrode reaction (initially at equilibrium) involving uncharged species, at a planar electrode in the absence of the supporting electrolyte; and current step chronopotentiometry for a liquid|liquid junction between two binary electrolytes. The calculations lead to analogous conclusions, with regard to the performance of the strategy, as in the case of the models represented by the Nernst–Planck–Poisson equations solved in Part 14. New observations reveal that highly non-linear boundary conditions present an additional challenge for the Newton iterative method utilised by the strategy. The ability of the strategy to solve non-local boundary conditions expressing current equality at distant boundaries is also demonstrated. However, formulating such boundary conditions correctly is not straightforward, so that their use is not recommended. Adaptive solutions suggest inaccuracies in the literature solutions for two example models.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 14: extension of the patch-adaptive strategy to time-dependent models involving migration–diffusion transport in one-dimensional space geometry, and its application to example transient experiments described by Nernst–Planck–Poisson equations

The finite-difference patch-adaptive strategy for electrochemical kinetic simulations, introduced in Part 5 and extended in Part 10 of this series of papers, is further extended to time-dependent models involving migration–diffusion transport in one-dimensional space geometry. The extensions include: spatial discretisation of generalised second spatial derivative expressions typical of migration–diffusion equations; allowance for the dependence of boundary conditions on displacement current; support for an a posteriori calculation of the displacement current as one of the model responses; optional calculation of steady-state initial conditions; the ability to use enhanced precision of floating point calculations. The extended strategy is used to simulate four examples of transient experiments, represented by Nernst–Planck–Poisson equation systems: coulostatic charge injection at an ideally polarised planar electrode; a voltage step for a thin layer asymmetric electrochemical cell; chronopotentiometry for an electrolyte|membrane|electrolyte system; and chronopotentiometry for a bipolar membrane. The strategy provides fairly reliable solutions, but its automatism and efficiency are less satisfactory compared to models without electric migration, owing to the need for model-dependent tuning of the method parameters, and increased computational cost necessary for the exact adaptive determination of the electric potential profiles.

An adaptive solution of the 3-D Euler equations on an unstructured grid

A new algorithm to generate the unstructured grid on a curved surface is developed. The advancing front method is used to generate the tetrahedral meshes in the space. An adaptive grid technique is used to enhance the calculation efficiency. The AUSM+ (Advection Upstream Splitting Method) scheme which was developed on a structured grid has been extended to be used to the spatial discretization of a cell-centered finite volume formulation on the unstructured grid. A second order spatial accuracy is achieved by applying a novel cell reconstruction procedure which can prevent the solution from exhibiting spurious oscillations without adding a limiter. A 3-D Euler solver for an adaptive tetrahedral grid and numerical results for several cases are presented.

On Accuracy of Adaptive Grid Methods for Captured Shocks

The accuracy of two grid adaptation strategies, grid redistribution and local grid refinement, is examined by solving the 2-D Euler equations for the supersonic steady flow around a cylinder. Second- and fourth-order linear finite difference shock-capturing schemes, based on the Lax–Friedrichs flux splitting, are used to discretize the governing equations. The grid refinement study shows that for the second-order scheme, neither grid adaptation strategy improves the numerical solution accuracy compared to that calculated on a uniform grid with the same number of grid points. For the fourth-order scheme, the dominant first-order error component is reduced by the grid adaptation, while the design-order error component drastically increases because of the grid nonuniformity. As a result, both grid adaptation techniques improve the numerical solution accuracy only on the coarsest mesh or on very fine grids that are seldom found in practical applications because of the computational cost involved. Similar error behavior has been obtained for the pressure integral across the shock. A simple analysis shows that both grid adaptation strategies are not without penalties in the numerical solution accuracy. Based on these results, a new grid adaptation criterion for captured shocks is proposed.

Actaptive grid technique based on the variational principle and its weight functions

Two computational cases that have analytic solutions are employed for studying the adaptive grid technique based on the variational principle. The results show that for the computational case of traveling shock waves the weight function, with the 2nd-order derivation terms taken into consideration, can more effectively reduce the error than one with gradient terms. For the case of cyclonic frontogenesis, weight functions only related to the gradient are unable to enhance the computational accuracy while ones with the wind field and frontogenesis function taken into consideration can more reasonably arrange the grid. Compared with analytic solutions, the adaptive grid technique suggested in this paper can improve computational accuracy and it displays the prominent advantage of saving memory.

An Adaptive Grid, Implicit Code for Spherically Symmetric, General Relativistic Hydrodynamics in Comoving Coordinates

We describe an implicit general relativistic hydrodynamics code. The evolution equations are formulated in comoving coordinates. A conservative finite differencing of the Einstein equations is outlined, and artificial viscosity and numerical diffusion are discussed. The time integration is performed with AGILE, an implicit solver for stiff algebrodifferential equations on a dynamical adaptive grid. We extend the adaptive grid technique, known from nonrelativistic hydrodynamics, to the general relativistic application and identify it with the concept of shift vectors in a 3+1 decomposition. The adaptive grid minimizes the number of required computational zones without compromising the resolution in physically important regions. Thus, the computational effort is greatly reduced when the zones are subject to computationally expensive additional processes, such as Boltzmann radiation transport or a nuclear reaction network. We present accurate results in the standard tests for supernova simulations: Sedov's point-blast explosion, the nonrelativistic and relativistic shock tube, the Oppenheimer-Snyder dust collapse, and homologous collapse.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations.: Part 13. Patch-adaptive simulation of wave propagation along ring electrodes: one-dimensional approximation

The patch-adaptive strategy for electrochemical kinetic simulations, introduced in Part 5 of this series of papers, is used to simulate the propagation of potential and concentration waves along ring electrodes, under galvanostatic control. The simulation is based on the one-dimensional model of Koper and Sluyters, combined with periodic boundary conditions adequate for ring electrodes. In this way, the extension of the strategy to non-local boundary conditions, described in Part 10 of this series of papers, is tested in the presence of a difficult-to-solve moving front crossing the boundaries with the periodic boundary conditions. The validity of the strategy for simulating this kind of pattern formation at electrodes is confirmed by comparisons with analytical solutions for the asymptotic case of the model. The adaptive simulations reveal that close to the asymptotic conditions, the initial, fast relaxation of the propagating waves of the electric potential drop across the double layer, does not immediately result in a steady-state, but it is followed by a slow decay of the remaining, non-moving spatial inhomogeneities of the concentration and potential profiles along the ring electrode.

Numerical Simulation of Shock Wave Reflecting Patterns for Different Flow Conditions

The numerical experiment has been conducted to investigate the unsteady shock wave reflecting phenomena. The cell-vertex finite-volume, Roe's upwind flux difference splitting method with unstructured grid is implemented to solve unsteady Euler equations. The <TEX>$4^{th}$</TEX>-order Runge-Kutta method is applied for time integration. A linear reconstruction of the flux vector using the least-square method is applied to obtain the <TEX>$2^{nd}$</TEX>-order accuracy for the spatial derivatives. For a better resolution of the shock wave and slipline, the dynamic grid adaptation technique is adopted. The new concept of grid adaptation technique, which is much simpler than that of conventional techniques, is introduced for the current study. Three error indicators (divergence and curl of velocity, and gradient of density) are used for the grid adaptation procedure. Considering the quality of the solution and the numerical efficiency, the grid adaptation procedure was updated up to <TEX>$2^{nd}$</TEX> level at every 20 time steps. For the convenience of comparison with other experimental and analytical results, the case of interaction between the straight incoming shock wave and a sharp wedge is simulated for various flow conditions. The numerical results show good agreement with other experimental and analytical results, in the shock wave reflecting structure, slipline, and the trajectory of the triple points. Some critical cases show disagreement with the analytical results, but these cases also have been proven to show hysteresis phenomena.

Ultra high pressure (UHP) terrains: Lessons from thermal modeling

The thermal history of subducted/exhumed crustal fragments is explored to depths of 125 km using a forward modeling approach. The finite element code uses an adaptive gridding technique that allows deformation and differential movement of subgrids to simulate tectonic mass flow. The geometry and velocity of plate convergence and the large-scale strain partitioning (for example, subduction channel, mobile tectonic fragments) are described explicitly as functions of time. The evolution of metamorphic conditions during descent and subsequent exhumation of a terrane within the subduction channel is computed for different kinematic scenarios. These include a range of velocities, relative to a stable continental block, for the subducting slab, the exhuming (UHP) fragment, and a subduction channel of given width. Though essentially generic in emphasis, our models are designed to help understand real UHP occurrences, most especially their exhumation history. Boundary conditions and the geometric set-up were chosen according to three case studies of known UHP terranes (eastern China, Western Norway, Western Alps). Simulations indicate that the temperature reached at a given depth during the subduction stage is crucially dependent on the convergence rate, width of the subduction channel, location of the fragment within the subduction channel, and on rates of shear heating and convection of the subcontinental mantle. The P-T-t path experienced by a tectonic fragment during its exhumation from Pmax is determined chiefly by its size (thickness) and exhumation velocity; the rate of ongoing subduction is less relevant given the rapid exhumation required to preserve UHP assemblages. The depth at which Tmax is reached by an exhuming UHP fragment and the dP/dT slope of its exhumation path may critically affect the extent of irreversible phase transitions during decompression. Comparing data for the three UHP case studies with corresponding simulation results indicates general agreement of the P-T paths. Predicted T-t evolution paths are also compatible with well-constrained mineral chronometry from the study areas. This agreement between field data and simulation results leads to a number of observables that may be useful in relating UHP occurrences to their evolution. For example, no metamorphic field gradients are expected for fragments of less than 5 km thickness, and the spread in mineral ages reflecting the high-pressure stage is predicted to be less than 1 Ma. Therefore, observed variations in metamorphic conditions for the stage near Tmax and P(Tmax) may reflect real differences of evolution in very large fragments only; and the same is valid for a substantial spread in metamorphic age data pertaining to stages near Pmax or Tmax. Tectonic extrusion of fragments along a subduction channel is not the only mechanism proposed to explain UHP occurrences. However, the broad agreement between several test cases and the present simulation results lend support to the viability of the kinematic scenario adopted. Where field evidence indicates a similar emplacement mechanism and allows the (initial) size of the UHP fragment to be inferred, the MELONPIT model provides a powerful integrative tool to produce a quantitative picture of the dynamics of UHP terranes.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 12. Patch-adaptive simulation of example transient experiments described by kinetic models defined over multiple space intervals in one-dimensional space geometry

The patch-adaptive strategy for electrochemical kinetic simulations, with extensions described in Part 10, is applied to three representative examples of time-dependent kinetic models defined over multiple space intervals in one-dimensional space geometry. The models describe: cyclic voltammetry for a diffusion-controlled electron transfer reaction at a liquid ∣ liquid interface; linear potential sweep voltammetry for a diffusion-controlled charge transfer at a planar amalgam film electrode; and potential step chronoamperometry for a free monoenzymatic amperometric biosensor. The strategy provides accurate, efficient and nearly automatic solutions for these examples, which confirms the overall adequacy of the extensions of the strategy, associated with the handling of multiple space intervals. However, limiting cases of the models, characterised by disparate lengths of the spatial intervals, appear difficult to solve. A possible occurrence of an error in the literature simulation results for the biosensor example is also revealed.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 11. Patch-adaptive simulation of example transient experiments described by kinetic models involving simultaneously distributed and localised unknowns, in one-dimensional space geometry

The patch-adaptive strategy for electrochemical kinetic simulations, with extensions described in Part 10, is applied to three representative examples of time-dependent kinetic models in one-dimensional space geometry, characterised by the simultaneous presence of spatially distributed and localised unknowns. The models describe: potential step chronoamperometry for a diffusion-controlled charge transfer reaction accompanied by the adsorption of reactants at the electrode; cyclic voltammetry for an electrochemical deposition–precipitation mechanism; and double potential step chronoamperometry for an oscillatory system involving a diffusion-controlled charge transfer reaction having a negative polarisation resistance, in the presence of double layer charging, and ohmic potential drops. The strategy provides accurate, efficient and nearly automatic solutions for these examples, which confirms the overall adequacy of the extensions of the strategy, associated with the handling of localised unknowns. In the third example, the adaptive simulations are very detailed and informative. They confirm the occurrence of numerical artefacts in the literature results of fixed grid simulations for this example.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 10. Extension of the patch-adaptive strategy to kinetic models involving spatially localised unknowns at the boundaries, multiple space intervals, and non-local boundary conditions, in one-dimensional space geometry

The finite-difference patch-adaptive strategy for electrochemical kinetic simulations, described in Part 5 of this series of papers, is extended to time-dependent models in one-dimensional space geometry, involving spatially localised unknowns at the boundaries, multiple space intervals, and non-local boundary conditions. These extensions are of interest, among others, for the modelling studies of: electrochemical adsorption and/or electrocatalytic reactions, liquid ∣ liquid interfaces, amalgam electrodes, electrochemical biosensors, and pattern formation at electrodes. The following new features are included in the strategy. Grid objects, used by the spatial grid adaptation algorithm, are decomposed into interior and boundary subgrid objects, keeping discrete values of the spatially distributed and localised unknowns, respectively. Linear algebraic equation systems, characterised by quasi-block-tridiagonal matrices having variable block dimensions, and isolated block-pentadiagonal rows, are solved by a specially developed variant of the Thomas algorithm, previously published by the present author. Relative error control is applied to localised unknowns. The calculation of consistent boundary solutions takes into account temporal derivatives present in boundary conditions. Accurate a posteriori calculation of temporal derivatives of the localised unknowns is enabled.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Advantage of time step adaptation, using example of current spikes in linear potential sweep voltammograms for the EqrevEqrev-DISP reaction mechanism

Electrochemical kinetic transients may exhibit brief, but rapid temporal variations, the location of which along the time axis, and duration, are difficult or impossible to predict. For example, very narrow current spikes may occur on linear potential sweep voltammograms for the E qrev E qrev-DISP reaction mechanism. Traditional fixed time grids require a dense grid spacing over large time intervals in such cases, which renders digital simulations computationally expensive. In contrast, the adaptive time step selection, included into the previously described finite-difference patch-adaptive strategy (PAS), enables a detailed temporal resolution of the current spikes, simultaneously with an efficient simulation of large-scale temporal variations. Using this technique, the phenomenon of the current spikes is investigated.

The application of a conservative grid adaptation technique to 1D shallow water equations

In this contribution, we present an efficient conservative mesh adaptation algorithm applied to 1D shallow water equations. This algorithm is suitable for unsteady situations and discontinuities of the solutions are well captured. Numerical results are presented.

A 2D fluid model of the scrape-off layer (SOL) using adaptive unstructured finite volumes

Recent advances in the development of adaptive grid methods for the 2D fluid simulation of tokamak edge plasmas are presented. Based on previous experiences on the solution of scalar model problems using both finite elements (FE) and finite volumes (FV), the first extension of the technique to a simplified version of the full system of Braginskii equations is presented here. The performances of adaptive grid technique are discussed and compared to the traditional fixed meshes on model problems representing a steady-state detachment and a transient evolution.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Patch-adaptive simulation of moving fronts in non-linear diffusion models of the switching of conductive polymers

The finite-difference patch-adaptive strategy (PAS) for electrochemical kinetic simulations, previously described by the author, is extended and applied to two examples of non-linear diffusion in one-dimensional space geometry, characterised by moving fronts, and related to the modelling of redox switching of conductive polymers. The extension consists of an appropriate spatial discretisation of the second derivative diffusion term involving concentration-dependent diffusion coefficient, and of an improved selection of starting approximations for the Newton iterations within the extrapolation time-stepping scheme. Adaptive solutions obtained are reliable, accurate, efficient, and superior to fixed-grid solutions.

Quantum wave packet dynamics with trajectories: Implementation with adaptive Lagrangian grids

The quantum trajectory method was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for fluid elements (“particles”) moving under the influence of the combined force from the potential surface and the quantum potential. To accurately compute the quantum potential and the quantum force, it is necessary to obtain the derivatives of a function given only the values on the unstructured mesh defined by the particle locations. However, in some regions of space–time, the particle mesh shows compression and inflation associated with regions of large and small density, respectively. Inflation is especially severe near nodes in the wave function. In order to circumvent problems associated with highly nonuniform grids defined by the particle locations, adaptation of moving grids is introduced in this study. By changing the representation of the wave function in these local regions (which can be identified by diagnostic tools), propagation is possible to much longer times. These grid adaptation techniques are applied to the reflected portion of a wave packet scattering from an Eckart potential.

Review of FDTD time-stepping schemes for efficient simulation of electric conductive media

We review the time-backward (TB), the time-forward (TF), and time-average (TA), and the exponential time-differencing (ETD) time-stepping schemes of the finite-difference time-domain (FDTD) method for the efficient simulation of electric conductive media. The application of global time-step adjustment and adaptive-gridding techniques is addressed in an outlook. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 25: 16–21, 2000.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations

The patch-adaptive strategy described in Part 5 proves to provide correct, fairly efficient, and nearly automatic solutions to six representative example electrochemical kinetic models in one-dimensional space geometry, that exhibit difficult-to-resolve local spatial and temporal solution structures at the electrodes. The models describe: the square-wave controlled potential transient for an uncomplicated heterogeneous charge transfer reaction, the potential step method and linear potential sweep voltammetry for the standard catalytic reaction mechanism with a pseudo-first-order homogeneous reaction, linear potential sweep voltammetry for an EC reaction mechanism involving a follow-up homogeneous dimerisation, steady-state voltammetry at a rotating disc electrode, and steady-state voltammetry at a spherical microelectrode. These examples are characterised by extremely thin reaction layers and other instances of local spatial regions of considerable variations of the solution, including transient effects caused by temporal discontinuities of boundary conditions. The strategy creates dynamically spatio-temporal grids well adapted to such solution structures, without any a priori knowledge about their location. The accuracy of the numerical solutions proves well controlled by means of error tolerance parameters. Numerical difficulties observed with the previously described adaptive moving grid technique do not occur in the present strategy. However, further work is needed to improve the performance of the strategy in the cases of thin reaction layers associated with fast second-order homogeneous reactions, and boundary layers associated with hydrodynamic electroanalytical methods, for which the computational effort appears too large.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations

Abstract Local, difficult-to-solve spatio-temporal structures such as extremely thin reaction layers at the electrodes, thin moving reaction fronts, fast transient variations, isolated temporal discontinuities in boundary conditions, edge effects, etc. are characteristic of the initial boundary value and boundary value problems occurring in electrochemical kinetic modelling. Further progress in the simulation methodology is hindered by the lack of automatic solution techniques for such problems. Based on a critical consideration of various adaptive grid strategies for partial differential equations reported in the scientific literature, a new finite-difference adaptive grid strategy has been formulated, that is especially designed for electrochemical kinetic simulations, and is currently limited to models in one-dimensional space geometry. The intention is to enable an automatic solution of the governing partial/ordinary differential equations to a prescribed accuracy, without any a priori knowledge about the spatio-temporal location of the emerging solution structures. In view of the importance of the solution (concentration) gradients for the electrochemical theory, simultaneous control of the spatial errors of the solutions and their spatial gradients is included in the strategy. Spatial grid adaptation is based on a local uniform grid refinement, using overlapping grid patches. Temporal step selection uses a recent control theoretic algorithm, combined with a simple method of detecting temporal discontinuities of the boundary conditions. A third-order accurate, implicit and L-stable Rosenbrock time-stepping scheme ROWDA3 is used to enable efficient and non-oscillatory temporal integration in the time intervals where a regular solution occurs. A second-order accurate Lawson–Morris–Gourlay extrapolation scheme is used to enable a reliable time-stepping at temporal discontinuities.