Bashforth Scheme Research Articles

Jump-diffusion models with two stochastic factors for pricing swing options in electricity markets with partial-integro differential equations

In this paper we consider the valuation of swing options with the possibility of incorporating spikes in the underlying electricity price. This kind of contracts are modelled as path dependent options with multiple exercise rights. From the mathematical point of view the valuation of these products is posed as a sequence of free boundary problems where two consecutive exercise rights are separated by a time period. Due to the presence of jumps, the complementarity problems are associated with a partial-integro differential operator. In order to solve the pricing problem, we propose appropriate numerical methods based on a Crank–Nicolson semi-Lagrangian method for the time discretization of the differential part of the operator, jointly with the explicit treatment of the integral term by using the Adams–Bashforth scheme and combined with biquadratic Lagrange finite elements for space discretization. In addition, we use an augmented Lagrangian active set method to cope with the early exercise feature. Moreover, we employ appropriate artificial boundary conditions to treat the unbounded domain numerically. Finally, we present some numerical results in order to illustrate the proper behaviour of the numerical schemes.

Mathematical modeling of electron irradiation of oil

PurposeThis paper aims to propose a mathematical model and numerical modeling to study the behavior of low conductive incompressible multicomponent hydrocarbon mixture in a channel under the influence of electron irradiation. In addition, it also aims to present additional mechanisms to study the radiation transfer and the separation of the mixture’s components.Design/methodology/approachThe three-dimensional non-stationary Navier–Stokes equation is the basis for this model. The Adams–Bashforth scheme is used to solve the convective terms of the equation of motion using a fourth-order accuracy five-point elimination method, and the viscous terms are computed with the Crank–Nicolson method. The Poisson equation is solved with the matrix sweep method and the concentration and electron irradiation equations are solved with the Crank–Nicolson method too.FindingsIt shows high computational efficiency and good estimation quality. On the basis of numerical results of mathematical model, the effect of the separation of mixture to fractions with various physical characteristics was obtained. The obtained results contribute to the improvement of technologies for obtaining high-quality oil products through oil separation into light and heavy fractions. Mathematical model is approbated based on test problem, and has good agreement with the experimental data.Originality/valueThe constructed mathematical model makes developing a methodology for conducting experimental studies of this phenomenon possible.

The decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations with nonsmooth initial data

In this paper, the decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations is considered with nonsmooth initial data. Our numerical scheme is based on the implicit Crank–Nicolson scheme for the linear terms and the explicit Adams–Bashforth scheme for the nonlinear terms for the temporal discretization, standard Galerkin finite element method is used to the spatial discretization. In order to improve the computational efficiency, the decoupled method is introduced, as a consequence the original problem is split into two linear subproblems, and these subproblems can be solved in parallel. We verify that our numerical scheme is almost unconditionally stable for the nonsmooth initial data (u0, θ0) with the divergence-free condition. Furthermore, under some stability conditions, we show that the error estimates for velocity and temperature in L2 norm is of the order O(h2+Δt32), in H1 norm is of the order O(h2+Δt), and the error estimate for pressure in a certain norm is of the order O(h2+Δt). Finally, some numerical examples are provided to verify the established theoretical findings and test the performances of the developed numerical scheme.

Decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations with smooth initial data

ABSTRACTBased on the mixed finite element method, we consider the decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations with smooth initial data in this paper. The temporal treatment of the spatial discrete Boussinesq equations is based on the implicit Crank–Nicolson scheme for the linear terms and the explicit Adams–Bashforth scheme for the nonlinear terms. Thanks to the decoupled method, the considered problem is split into two subproblems and these subproblems can be solved in parallel. Under some restriction on the time step, we present the stability and convergence results of numerical solutions, Finally, some numerical experiments are provided to test the performance of the developed numerical scheme and verify the established theoretical findings.

Convergence of an implicit–explicit midpoint scheme for computational micromagnetics

Based on lowest-order finite elements in space, we consider the numerical integration of the Landau–Lifschitz–Gilbert equation (LLG). The dynamics of LLG is driven by the so-called effective field which usually consists of the exchange field, the external field, and lower-order contributions such as the stray field. The latter requires the solution of an additional partial differential equation in full space. Following Bartels and Prohl (2006), we employ the implicit midpoint rule to treat the exchange field. However, in order to treat the lower-order terms effectively, we combine the midpoint rule with an explicit Adams–Bashforth scheme. The resulting integrator is formally of second-order in time, and we prove unconditional convergence towards a weak solution of LLG. Numerical experiments underpin the theoretical findings.

Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative

Recently a new fractional differentiation was introduced to get rid of the singularity in the Riemann-Liouville and Caputo fractional derivative. The new fractional derivative has then generate a new class of ordinary differential equations. These class of fractional ordinary differential equations cannot be solved using conventional Adams–Bashforth numerical scheme, thus, in this paper a new three-step fractional Adams–Bashforth scheme with the Caputo–Fabrizio derivative is formulated for the solution linear and nonlinear fractional differential equations. Stability analysis result shows that the proposed scheme is conditionally stable. Applicability and suitability of the scheme is justified when applied to solve some novel chaotic systems with fractional order α ∈ (0, 1).

The Crank–Nicolson/Adams–Bashforth scheme for the Burgers equation with H2 and H1 initial data

In this paper, we consider the stability and convergence results of the Crank–Nicolson/Adams–Bashforth scheme for the Burgers equation with smooth and nonsmooth initial data. The spatial approximation is based on the standard conforming finite element space. The temporal treatment of the spatial discrete Burgers equation is based on the implicit Crank–Nicolson scheme for the linear term and the explicit Adams–Bashforth scheme for the nonlinear term. Firstly, we prove that the Crank–Nicolson/Adams–Bashforth scheme is almost unconditionally stable with initial data u0∈Hα (α=1,2). Secondly, the optimal error estimates of the numerical solution in L2-norm are derived with initial data u0∈H2, and the error estimates of approximate solution in L2 norm obtained with initial data u0∈H1 is reduced by 12. Finally, some numerical examples are provided to verify the established stability theory and convergence results with H2 and H1 initial data.

Geometry modeling and permeability prediction for textile preforms with nesting in laminates

In the multilayer textile preform which is usually used in the resin transfer molding (RTM) process, the penetration phenomenon existing between two neighboring layers is referred to as nesting, which has important influence on the permeability of the preforms. Predicting the permeability of the textile preforms with nesting accurately is the foundation of mold‐filling simulation and process optimization for RTM. In this article, based on the analyses of the penetration and local deformation behaviors between two neighboring layers, an assumption on constant area of yarn cross‐section is established, and a long axis compensation method is proposed to build the geometry model of the multilayer textile preform with nesting in laminates. The governing equations of the resin flow in textile preform are built, and a high resolution total variation diminishing difference scheme is constructed based on Adams–Bashforth scheme and Chorin projection method to numerically solve the above equations and predict the permeability of the preform. According to the comparison between simulation predictions and experimental measurements, the validities of the above geometry modeling method and permeability prediction algorithm are proved. POLYM. COMPOS., 39:4408–4415, 2018. © 2017 Society of Plastics Engineers

Implementation of Delayed Detached Eddy Simulation method to a high order spectral difference solver

Abstract The Delayed Detached Eddy Simulation (DDES) method based on Spalart–Allmaras (SA) turbulence model is implemented to a high order spectral difference solver for aeroacoustics problems in this study. A modified SA model by Crivellini et al. is used to eliminate the instability of the nonlinear source terms in the turbulence model. To speed up the unsteady simulation, the multi-time-step method based on the optimized Adams–Bashforth scheme is utilized for time marching. The perfectly matched layer absorbing boundary condition for Navier–Stokes equations is applied at the far-field and outflow boundary regions. The SA model is firstly validated with the high Reynolds number flow over a flat plate and the flow around a NACA0012 airfoil at 10° and 15° angles of attack. The numerical results are compared with the experimental or reference data by other researchers. The good agreement indicates that the high order spectral difference solver with SA model is accurate to compute the turbulence boundary layer. Then the developed DDES solver is validated with the backward facing step flow and the flow over a NACA0012 airfoil at 60° angle of attack. The computed results are compared with the experimental data, as well as the URANS results. The validation results show that the DDES solver is capable of computing the turbulence flow with boundary layer separations. Finally the noise from the flow past two cylinders in tandem is computed with the DDES solver. The computed results, including the time mean flow field, dynamic pressures on both cylinders and far field noise spectra are compared thoroughly with the experimental data provided by NASA, as well as the reference numerical results by other researchers. An assessment is made of the accuracy of the present high order SD solver versus the low order method. The results show that the high order SD solver with the DDES method is more accurate than low order solver for aeroacoustics numerical simulation with similar resolution, and capable of predicting accurately the noise for aeroacoustics problems with turbulence separations.

Modelling uncertainty in incompressible flow simulation using Galerkin based generalized ANOVA

This paper presents a new algorithm, referred to here as Galerkin based generalized analysis of variance decomposition (GG-ANOVA) for modelling input uncertainties and its propagation in incompressible fluid flow. The proposed approach utilizes ANOVA to represent the unknown stochastic response. Further, the unknown component functions of ANOVA are represented using the generalized polynomial chaos expansion (PCE). The resulting functional form obtained by coupling the ANOVA and PCE is substituted into the stochastic Navier–Stokes equation (NSE) and Galerkin projection is employed to decompose it into a set of coupled deterministic ‘Navier–Stokes alike’ equations. Temporal discretization of the set of coupled deterministic equations is performed by employing Adams–Bashforth scheme for convective term and Crank–Nicolson scheme for diffusion term. Spatial discretization is performed by employing finite difference scheme. Implementation of the proposed approach has been illustrated by two examples. In the first example, a stochastic ordinary differential equation has been considered. This example illustrates the performance of proposed approach with change in nature of random variable. Furthermore, convergence characteristics of GG-ANOVA has also been demonstrated. The second example investigates flow through a micro channel. Two case studies, namely the stochastic Kelvin–Helmholtz instability and stochastic vortex dipole, have been investigated. For all the problems results obtained using GG-ANOVA are in excellent agreement with benchmark solutions.

Analytical and numerical solutions of the generalized dispersive Swift–Hohenberg equation

The generalization of the Swift–Hohenberg equation is studied. It is shown that the equation does not pass the Kovalevskaya test and does not possess the Painlevé property. Exact solutions of the generalized Swift–Hohenberg equation which are very useful to test numerical algorithms for various boundary value problems are obtained. The numerical algorithm which is based on the Crank–Nicolson–Adams–Bashforth scheme is developed. This algorithm is tested using the exact solutions. The selforganization processes described by the generalization of the Swift–Hohenberg equation are studied.

Second order time–space iterative method for the stationary Navier–Stokes equations

A second order time–space implicit/explicit iterative scheme for the stationary Navier–Stokes equations is designed, where the spatial discretization is based on the mixed finite element method and the time discretization is based on the second order implicit/explicit (the Crank–Nicolson/Admas–Bashforth) scheme. Under a weak uniqueness condition, the optimal H 1 - L 2 error estimates related to the mesh size h and time step size τ of the iterative solution ( u h n , p h n ) to the exact solution ( u ̃ , p ̃ ) and the optimal L 2 error estimate related to h and τ of the iterative solution u h n to the exact solution u ̃ are provided. In numerical aspect, some comparisons with the first order time–space iterative method are made to confirm the efficiency of the proposed second order scheme.

A GPU-accelerated adaptive discontinuous Galerkin method for level set equation

ABSTRACTThis paper presents a GPU-accelerated nodal discontinuous Galerkin method for the solution of two- and three-dimensional level set (LS) equation on unstructured adaptive meshes. Using adaptive mesh refinement, computations are localised mostly near the interface location to reduce the computational cost. Small global time step size resulting from the local adaptivity is avoided by local time-stepping based on a multi-rate Adams–Bashforth scheme. Platform independence of the solver is achieved with an extensible multi-threading programming API that allows runtime selection of different computing devices (GPU and CPU) and different threading interfaces (CUDA, OpenCL and OpenMP). Overall, a highly scalable, accurate and mass conservative numerical scheme that preserves the simplicity of LS formulation is obtained. Efficiency, performance and local high-order accuracy of the method are demonstrated through distinct numerical test cases.

Stability of the Crank–Nicolson–Adams–Bashforth scheme for the 2D Leray‐alpha model

We consider the stability of an efficient Crank–Nicolson–Adams–Bashforth method in time, finite element in space, discretization of the Leray‐α model. We prove finite‐time stability of the scheme in L2, H1, and H2, as well as the long‐time L‐stability of the scheme under a Courant‐Freidrichs‐Lewy (CFL)‐type condition. Numerical experiments are given that are in agreement with the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1155–1183, 2016

An efficient and long-time accurate third-order algorithm for the Stokes–Darcy system

A third-order in time numerical IMEX-type algorithm for the Stokes---Darcy system for flows in fluid saturated karst aquifers is proposed and analyzed. A novel third-order Adams---Moulton scheme is used for the discretization of the dissipative term whereas a third-order explicit Adams---Bashforth scheme is used for the time discretization of the interface term that couples the Stokes and Darcy components. The scheme is efficient in the sense that one needs to solve, at each time step, decoupled Stokes and Darcy problems. Therefore, legacy Stokes and Darcy solvers can be applied in parallel. The scheme is also unconditionally stable and, with a mild time-step restriction, long-time accurate in the sense that the error is bounded uniformly in time. Numerical experiments are used to illustrate the theoretical results. To the authors' knowledge, the novel algorithm is the first third-order accurate numerical scheme for the Stokes---Darcy system possessing its favorable efficiency, stability, and accuracy properties.

Mixed stabilized finite element methods based on backward difference/Adams–Bashforth scheme for the time-dependent variable density incompressible flows

In this paper, we consider a second-order mixed stabilized finite element method based on pressure projection method for variable density incompressible flows. The originality of the proposed approach is to use a stabilized method based on the difference between a consistent and an under-integrated mass matrix of the pressure for variable density incompressible flows approximated by the lowest equal-order finite element pairs. A second-order backward difference (BDF) for the temporal term and a second-order Adams–Bashforth (AB) for the explicit treatment of the nonlinear term lead to the presented second-order BDF/AB scheme. The stability of the method was proved and the performance of the method is numerically illustrated. Finally, comparison with some established methods, a series of numerical experiments are given to show that this method has better stability and accuracy.

An efficient and accurate fully discrete finite element method for unsteady incompressible Oldroyd fluids with large time step

SummaryThis paper proposes a second‐order accuracy in time fully discrete finite element method for the Oldroyd fluids of order one. This new approach is based on a finite element approximation for the space discretization, the Crank–Nicolson/Adams–Bashforth scheme for the time discretization and the trapezoid rule for the integral term discretization. It reduces the nonlinear equations to almost unconditionally stable and convergent systems of linear equations that can be solved efficiently and accurately. Here, the numerical simulations for L2, H1 error estimates of the velocity and L2 error estimates of the pressure at different values of viscoelastic viscosities α, different values of relaxation time λ1, different values of null viscosity coefficient μ0 are shown. In addition, two benchmark problems of Oldroyd fluids with different solvent viscosity μ and different relaxation time λ1 are simulated. All numerical results perfectly match with the theoretical analysis and show that the developed approach gives a high accuracy to simulate the Oldroyd fluids under a large time step. Furthermore, the difference and the connection between the Newton fluids and the viscoelastic Oldroyd fluids are displayed. Copyright © 2015 John Wiley & Sons, Ltd.

A compact finite differences exact projection method for the Navier–Stokes equations on a staggered grid with fourth-order spatial precision

An exact projection method for the numerical solution of the incompressible Navier–Stokes equations is devised. In all spatial discretizations, fourth-order compact finite differences are used, including domain boundaries and the Poisson equation that arises from the projection method. The integration in time is carried out by a second-order Adams–Bashforth scheme. The discrete incompressibility constraint is imposed exactly (up to machine precision) by a simple and efficient discretization of the Poisson equation. Spatial and temporal accuracies, for both velocity and pressure, are verified through the use of analytical and manufactured solutions. The results show that the method converges with fourth-order accuracy in space and second-order accuracy in time, for both velocity and pressure. Additionally, two popular benchmark problems, the flow over a backward facing step and the lid-driven cavity flow, are used to demonstrate the robustness and correctness of the code.

Analysis of triangular C-grid finite volume scheme for shallow water flows

In this paper, a dispersion relation analysis is employed to investigate the finite volume triangular C-grid formulation for two-dimensional shallow-water equations. In addition, two proposed combinations of time-stepping methods with the C-grid spatial discretization are investigated. In the first part of this study, the C-grid spatial discretization scheme is assessed, and in the second part, fully discrete schemes are analyzed. Analysis of the semi-discretized scheme (i.e. only spatial discretization) shows that there is no damping associated with the spatial C-grid scheme, and its phase speed behavior is also acceptable for long and intermediate waves. The analytical dispersion analysis after considering the effect of time discretization shows that the Leap-Frog time stepping technique can improve the phase speed behavior of the numerical method; however it could not damp the shorter decelerated waves. The Adams–Bashforth technique leads to slower propagation of short and intermediate waves and it damps those waves with a slower propagating speed. The numerical solutions of various test problems also conform and are in good agreement with the analytical dispersion analysis. They also indicate that the Adams–Bashforth scheme exhibits faster convergence and more accurate results, respectively, when the spatial and temporal step size decreases. However, the Leap-Frog scheme is more stable with higher CFL numbers.

Numerical simulation of the three dimensional Allen–Cahn equation by the high-order compact ADI method

In this paper, a new linearized high-order compact difference method is presented for numerical simulation of three dimensional (3D) Allen–Cahn equation with three kinds of boundary conditions. The method, which is based on the Crank–Nicholson/Adams–Bashforth scheme combined with the Douglas–Gunn ADI method, is second order accurate in time and fourth order accurate in space and energy degradation. The main advantages of this method is that the nonlinear penalty term f(u) is linear and an extra stabilizing term is added to alleviate the stability constraint while maintaining accuracy and simplicity. Numerical experiments are given to demonstrate the validity and applicability of the new method.