Richardson Extrapolation Research Articles

Optimally accurate second-order numerical scheme on layer-resolving graded mesh for singularly-perturbed boundary turning point problems exhibiting power-of-type-1 and hybrid layers

The paper discusses singularly-perturbed problems with boundary turning points. It describes constructions of layer-resolving grids based on layer-eliminating coordinate transformations and analyses the convergence of numerical solutions obtained using the upwind scheme on the layer-resolving grids and Richardson extrapolations. Theoretical error estimates and numerical experiments confirm that the upwind scheme with Richardson extrapolation on the proposed layer-resolving grids is uniformly convergent of order two without any logarithmic factor.

A Benchmark for Neutron Kinetics Methods in Hexagonal Geometry

A new benchmark solution has been developed to aid in the development of neutron kinetics solvers for hexagonal geometries, such as those in water-water energetic reactors. This benchmark problem is based on the two-dimensional, two-group, International Atomic Energy Agency–Hex steady-state benchmark problem. Two transient problems are presented: a ramp and a step transient. To create a benchmark-quality solution to this transient problem, a basic neutron kinetics model was added to the computer program LUPINE (Liquid metal–cooled fast reactor Utility for Physics Informed Nuclear Engineering). LUPINE solves neutron kinetics equations in general unstructured mesh. First, the LUPINE kinetics solvers are verified using the TWIGL benchmark problems. Then the methods in LUPINE are used to perform a spatiotemporal convergence analysis to ensure that the solutions are sufficiently converged. Finally, Richardson extrapolation is performed to obtain the reference solutions for these new kinetics benchmark problems.

Erlangization/Canadization of Phase-Type Jump Diffusions, with Applications to Barrier Options

For a jump diffusion with upward phase-type jumps with p phases, the maximum before an independent Erlang time eηq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^q_\\eta$$\\end{document} with q stages and rate parameter η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta$$\\end{document} is again phase-type with q(p+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q(p+1)$$\\end{document} phases. An iterative scheme for computing the phase generator is presented and applied to representing the price of a barrier option with time horizon eηq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^q_\\eta$$\\end{document} as a single ordinary integral. Canadization then means to approximate a fixed horizon T with an eηq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^q_\\eta$$\\end{document} satisying Eeηq=T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {E}}e^q_\\eta =T$$\\end{document} for a sufficiently large q. Similar results holds for Greeks like the delta and the gamma. A numerical example is given for a down-and-in call option and the Canadization is combined with Richardson extrapolation. Finally, a recursion is developed that only requires the iteration to be performed in p+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p+1$$\\end{document} dimensions.

A sixth-order finite difference method for the two-dimensional nonlinear advection diffusion reaction equation

A novel higher-order finite difference method is developed to solve the two-dimensional nonlinear advection diffusion reaction equation. Firstly, a new five-point sixth-order finite difference scheme is proposed to discretize spatial derivatives and the sixth-order backward difference method is applied to teat temporal derivative. As a result, a new fully implicit scheme with sixth-order accuracy in both time and space is proposed. Subsequently, for the calculation of the starting step of the sixth-order backward difference method, the Crank-Nicolson method combined with the Richardson extrapolation technique is employed to obtain the equivalent temporal accuracy as the main scheme. Finally, the accuracy and reliability of the presented method are validated through various numerical examples.

The maximum norm error estimate and Richardson extrapolation methods of a second-order box scheme for a hyperbolic-difference equation with shifts

The study aims at the development and theoretical analyses of a box method used to solve a kind of hyperbolic equations with shifts. By using the discrete energy method, it is shown that numerical solutions converge to exact solutions with an order of O(τ 2 + h 2) in L ∞-norm as τ = O(h). Here, τ and h are temporal and spatial meshsizes, respectively. According to local truncation error, the asymptotic expansion formula of the numerical solutions is derived by introducing two auxiliary problems. By applying this asymptotic expansion formula, a class of Richardson extrapolations methods have been derived to achieve extrapolation solutions with an order of O(τ 4 + h 4) in L ∞-norm. Numerical results show the high performance of the proposed methods and the exactness of theoretical findings.

An efficient numerical approach for singularly perturbed time delayed parabolic problems with two-parameters

ObjectivesThe main objective of this work is to design an efficient numerical scheme is proposed for solving singularly perturbed time delayed parabolic problems with two parameters.ResultsThe scheme is constructed via the implicit Euler and non-standard finite difference method to approximate the time and space derivatives, respectively. Besides, to enhance the accuracy and order of convergence of the method Richardson extrapolation technique is employed. Intensive numerical experimentation has been done on some model examples. Further, the layer behavior of the solutions is presented using graphs and observed to agree with the existing theories. Finally, error analysis of the scheme is done and observed that the proposed method is parameter uniform convergent with the order of convergence (Δt)2+h2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( (\\Delta t )^2+h^2\\right) $$\\end{document}

Richardson Extrapolation of the Euler Scheme for Backward Stochastic Differential Equations

Richardson Extrapolation of the Euler Scheme for Backward Stochastic Differential Equations

A generalized differential quadrature approach to the modelling of heat transfer in non-similar flow with nonlinear convection

This study aims to investigate heat transfer and entropy production in the non-similar flow of an incompressible fluid, with nonlinear convection and viscous dissipation. To obtain precise solutions, the Sparrow-Quack-Boerner local non-similarity method is implemented. The non-similarity arises from the nonlinear convection term present in the momentum equation. The non-similarity arises from the nonlinear convection term present in the momentum equation. Consequently, the inclusion of non-similarity terms into the energy equation is achieved through the interconnection of momentum and energy equations. Entropy generation is explored by employing the second law of thermodynamics. Numerical results from several truncation levels are presented in tabular form, and equations for both first and second-level truncations are generated. The one- and two-equation models are solved by implementing the Generalized Differential Quadrature Method (GDQM). Comparison with a second level of truncation reveals significantly greater inaccuracies in numerical results obtained from the first level. This discrepancy arose due to the omission of non-similar terms in the primary governing equations at the first truncation stage. The validity and accuracy of the derived numerical solutions are further demonstrated by the application of the GDQM and the midpoint method with Richardson extrapolation. Additionally, a plot of the numerical data produced from the second level of truncations is presented together with a discussion of the various physical factors. Increasing the mixed convection parameter accelerates fluid velocity, also when the viscous dissipation parameter increases, temperature and the Bejan number increase. The Prandtl number and the thickness of the thermal boundary layer are observed to be inversely related. Moreover, the quantity of entropy generated at the stretching surface and inside the boundary layer rises in proportion to the increases in the Prandtl and Eckert numbers.

American put options with regime-switching volatility

We present an approach for pricing American put options with a regime-switching volatility. Our method reveals that the option price can be expressed as the sum of two components: the price of a European put option and the premium associated with the early exercise privilege. Our analysis demonstrates that, under these conditions, the perpetual put option consistently commands a higher price during periods of high volatility compared to those of low volatility. Moreover, we establish that the optimal exercise boundary is lower in high-volatility regimes than in low-volatility regimes. Additionally, we develop an analytical framework to describe American puts with an Erlang-distributed random-time horizon, which allows us to propose a numerical technique for approximating the value of American puts with finite expiry. We also show that a combined approach involving randomization and Richardson extrapolation can be a robust numerical algorithm for estimating American put prices with finite expiry.

Analysis of Mesh Resolution Effect to Numerical Result of CFD-ROM: Turbulent Flow in Stationary Parallel Plate

Computational fluid dynamics (CFD) is extensively utilized to predict flow behaviour in various industries and applications. The Full Order Model (FOM) is a high-accuracy approach to flow modelling, but it requires significant computational resources due to its high order and thousands of variables. To address this problem, the Reduced Order Model (ROM) was developed. Despite the advancement brought by ROM, there is a notable gap in research concerning the impact of mesh configuration on CFD-ROM results. While the number of modes has been extensively studied for its influence on CFD-ROM, the mesh configuration, a critical aspect of the simulation process, has received relatively limited attention. This study investigates the effect of mesh resolution on numerical results in CFD-ROM concerning turbulent flow within stationary parallel plates. Employing rigorous methods, including Richardson Extrapolation, verification, validation, and error percentage. The results explicitly confirm that mesh resolution directly impacts the numerical results of the velocity field in CFD-ROM. It is found that there is a notable reduction in Convergence Grid Index (CGI) values for different mesh ratios: 6.401% for medium-to-coarse and 2.031% for fine-to-medium ratio. Thus, with the same mode number, mesh resolution selection can enhance the numerical result of the velocity field in CFD-ROM.

The Boyle–Romberg Trinomial Tree, a Highly Efficient Method for Double Barrier Option Pricing

Oscillations in option price convergence have long been a problematic aspect of tree methods, inhibiting the use of repeated Richardson extrapolation that could otherwise greatly accelerate convergence, a feature integral to some of the most efficient modern methods. These oscillations are typically caused by the fluctuating positions of nodes around the discontinuities in the payoff function or its derivatives. Our paper addresses this crucial gap that typically prohibits the use of lattice methods when high efficiency is needed. Focusing on double barrier options, we develop a trinomial tree in which the positions of the nodes are precisely adjusted to align with these discontinuities throughout the option’s lifespan and across various time steps. This alignment enables the use of repeated extrapolation to achieve high order convergence, including near barriers, a well-known challenge in many tree methods. Maintaining the inherent simplicity and adaptability of tree methods, our approach is easily applicable to other models and option types.

Numerical Study of a Model and Full-Scale Container Ship Sailing in Regular Head Waves

In the present study, the added resistance, heave, and pitch of the KRISO Container Ship (KCS) in waves, at both model scale and full scale, are predicted numerically in regular head waves, for four wavelengths and three wave heights. The ISIS-CFD viscous flow solver, implemented in the Fidelity Fine Marine software provided by CADENCE, was employed for the numerical simulations. The spatial discretization was based on the finite volume method using an unstructured grid. The unsteady Reynolds-averaged Navier–Stokes (RANS) equations were solved numerically, with the turbulence modeled by shear stress transport (k-ω) (SST). The free-surface capturing was based on the volume-of-fluid method. The computed solutions were validated through comparisons with towing test data available in the public domain. To predict the uncertainties in the numerical solution, a systematic grid convergence study based on the Richardson extrapolation method was performed for a single wave case on three different grid resolutions. Specific attention was given to the free-surface and wake flow in the propeller plane. The purpose was to compare the numerical results from the model- and full-scale tests to examine the scale’s effect on the ship’s performance in regular head waves. The comparison between the model scale and full scale showed obvious differences, less accentuated for the free-surface topology and clearly observed in terms of boundary layer formation in the propeller’s vicinity.

A Novel Richardson Extrapolation Technique on a Shishkin and Exponentially Graded Mesh for Singularly Perturbed Convection-diffusion Problems With Integral Boundary Conditions

A Novel Richardson Extrapolation Technique on a Shishkin and Exponentially Graded Mesh for Singularly Perturbed Convection-diffusion Problems With Integral Boundary Conditions

Richardson extrapolation technique for singularly perturbed convection-diffusion problem with non-local boundary conditions

Richardson extrapolation technique for singularly perturbed convection-diffusion problem with non-local boundary conditions

Richardson Extrapolation Technique for Singularly Perturbed Parabolic Convection-Diffusion Problems With A Discontinuous Initial Condition

Richardson Extrapolation Technique for Singularly Perturbed Parabolic Convection-Diffusion Problems With A Discontinuous Initial Condition

Conservative higher-order finite difference scheme for the coupled nonlinear Schrödinger equations

This paper introduces a conservative higher-order finite difference scheme for solving the coupled nonlinear Schrödinger equations. The Crank–Nicolson method is employed to discretize time derivatives and the sixth-order difference operator is used to discretize space derivatives, correspondingly, the resulting difference scheme has second-order accuracy in time and sixth-order accuracy in space. By utilizing the discrete energy method, the conservation of discrete mass and energy, the boundedness, existence and uniqueness of solution, unconditional stability and the convergence of the new scheme are proved. Then making use of Richardson extrapolation, the time accuracy is increased to the fourth order. Finally, the numerical experiments are conducted to validate the theoretical results presented in the paper.

Accelerating Convergence in Trinomial Option Pricing: Recursive Incremental Value Ordering with Repeated Richardson Extrapolation

Accelerating Convergence in Trinomial Option Pricing: Recursive Incremental Value Ordering with Repeated Richardson Extrapolation

Verification of the γ-Reθt Transition Model in OVERFLOW and FUN3D

The findings of the recent AIAA and NATO workshops on transition modeling have underscored the importance of verifying the transition models that are based on Reynolds-averaged Navier–Stokes. To that end, the present work aims to verify the Langtry–Menter (LM2009) transition model coupled with Menter’s shear-stress transport (SST) turbulence model. Specifically, the SST-2003-LM2009 model, as implemented in the OVERFLOW structured overset solver and the FUN3D unstructured solver, is applied to both natural and bypass transition scenarios on a flat plate and to natural and separation-bubble-induced transition on the NLF-0416 airfoil. The Richardson extrapolation method was used to analyze and report the grid convergence of integrated and local load coefficients upstream, within, and downstream of the transition region, based on solutions obtained using the two solvers on identical grids. Reasonable agreement is obtained between the two codes for both global and local surface load coefficients when further refining the workshop mesh series, although with a grid-convergence index for the local friction coefficient that may exceed 1% in certain cases. The study also highlights the solution sensitivity to the choice of turbulence inflow boundary conditions and model options, along with the necessity to unambiguously specify this information for benchmark exercises.

Fitted mesh scheme for singularly perturbed parabolic convection–diffusion problem exhibiting twin boundary layers

In this paper, fitted mesh numerical scheme is presented for solving singularly perturbed parabolic convection–diffusion problem exhibiting twin boundary layers. To approximate the solution, we discretize the temporal variable on uniform mesh and discretize the spatial one on piecewise uniform mesh of the Shishkin mesh type. The resulting scheme is shown to be almost first order convergent that accelerated to almost second order convergent by applying the Richardson extrapolation technique. Stability and consistency of the proposed method are established very well in order to guarantee the convergence of the method. Further, the theoretical investigations are confirmed by numerical experiments. Moreover, the present scheme is stable, consistent and gives more accurate solution than existing methods in the literature.

Richardson Extrapolation Technique for Singularly Perturbed Convection-diffusion Problem with Non-local Boundary Conditions

This study presents the Richardson extrapolation techniques for solving singularly perturbed convection -diffusion problems (SPCDP) with non-local boundary conditions. A numerical approach is presented using an upwind finite difference scheme a piecewise-uniform (Shishkin) mesh. To handle the non-local boundary conditions, the trapezoidal rule is applied. The study establishes an error bound for numerical solutions and determines the numerical approximation for scaled derivatives. To enhance convergence and accuracy, we utilize Richardson extrapolation. This elevates accuracy from firs-order to second order convergence