Convergence Rate Of Order Research Articles

Discontinuous Galerkin methods using poly-sinc approximation

In this paper, we develop a discontinuous Galerkin method using poly-Sinc approximation, in which non-equidistant points are generated by conformal mappings, such as Sinc points. Sinc points are characterized by their dense distribution near the endpoints of an interval, which render them effective in dealing with jumps introduced by the partitioning process of the interval in a discontinuous Galerkin scheme. Sinc methods are characterized by their fast convergence rate of exponential order and effective handling of singularities. We use equidistant, Chebyshev, and Sinc partitions to handle the singularities. We demonstrate that, using poly-Sinc-based discontinuous Galerkin approximation, Sinc partitions outperforms those with equidistant and Chebyshev partitions as we near the singularity. We also demonstrate that for Sinc partitions, poly-Sinc-based discontinuous Galerkin approximation outperforms those with Chebyshev points. We used the stability condition of the discontinuous Galerkin method to demonstrate the stability of our approach. It was shown that the approximation error between a smooth function and its poly-Sinc approximation over the global partition has a convergence rate of exponential order. For a function with a singularity at an endpoint, we use the weighted L2 norm over the Sinc partitions. Using the weighted L2 norm, we show that, by computing the ℓ2 norm over all Sinc partitions constituting the global partition except the last partition, the approximation error between the exact solution of the ordinary differential equation and its poly-Sinc-based discontinuous Galerkin approximation has a convergence rate of exponential order.

Efficient Simulation of Stochastic Differential Equations Based on Markov Chain Approximations With Applications

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms \textit{both biased and unbiased} simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory. Note: this is an earlier version of the work Efficient Simulation of Generalized SABR and Stochastic Local Volatility Models based on Markov Chain Approximations.

A higher order scheme for singularly perturbed delay parabolic turning point problem

PurposeThe purpose of this study is to construct and analyze a parameter uniform higher-order scheme for singularly perturbed delay parabolic problem (SPDPP) of convection-diffusion type with a multiple interior turning point.Design/methodology/approachThe authors construct a higher-order numerical method comprised of a hybrid scheme on a generalized Shishkin mesh in space variable and the implicit Euler method on a uniform mesh in the time variable. The hybrid scheme is a combination of simple upwind scheme and the central difference scheme.FindingsThe proposed method has a convergence rate of order . Further, Richardson extrapolation is used to obtain convergence rate of order two in the time variable. The hybrid scheme accompanied with extrapolation is second-order convergent in time and almost second-order convergent in space up to a logarithmic factor.Originality/valueA class of SPDPPs of convection-diffusion type with a multiple interior turning point is studied in this paper. The exact solution of the considered class of problems exhibit two exponential boundary layers. The theoretical results are supported via conducting numerical experiments. The results obtained using the proposed scheme are also compared with the simple upwind scheme.

Approximation of Quasilinear Hyperbolic Problems with Discontinuous Coefficients: An Optimal Error Estimate

We develop a theoretical framework for the approximation of a class of second-order quasilinear hyperbolic interface problems on quadratic finite element. Time discretization based on linearized implicit difference scheme with degree two polynomials for interface approximation is proposed. Sufficient conditions, on the input data, that guarantee the existence of a unique solution are given. Under these assumptions, the stability of the scheme is established and convergence rate of optimal order is proved. It is assumed that the interface is arbitrary but smooth.

A priori error estimates of multiblock mortar expanded mixed method for elliptic problems

The multiblock mortar expanded mixed method is explored to solve the second order linear elliptic problem. In this method, the domain is expressed as a union of smaller blocks (subdomains) separated by interface. The original problem is posed on each block and discretized locally. A scalar unknown is introduced on the shared boundaries between the blocks which is treated as Dirichlet boundary condition for the local problem. A special finite element space is constructed on the interface which serve as Lagrange multiplier to impose flux continuity across the inter-block boundaries. The mortar space is also used to approximate the scalar variable introduced on interface. We applied expanded mixed method to solve the local problem on each block and derived the error estimates for scalar, its gradient and its flux. We established the optimal order convergence rates for subdomain approximations. An error bound for mortar pressure is also presented. The computation is performed by transforming the algebraic system encounter by formulation into the positive definite problem in mortar space. An algorithm illustrating the implementation of method is also provided. The numerical experiments confirming theoretical results are also provided.

Accurate solution-adaptive finite difference schemes for coarse and fine grids

We introduce solution dependent finite difference stencils whose coefficients adapt to the current numerical solution by minimizing the truncation error in the least squares sense. The resulting scheme has the resolution capacity of dispersion relation preserving difference stencils in under-resolved domains, together with the high order convergence rate of conventional central difference methods in well resolved regions. Numerical experiments reveal that the new stencils outperform their conventional counterparts on all grid resolutions from very coarse to very fine.

Stochastic Lohe Matrix Model on the Lie Group and Mean-Field Limit

We propose a Lohe matrix model in a random environment where each oscillator can be regarded as an element of a general matrix Lie group G. In order to make the stochastic system stays on G for all time, we introduce suitable noise terms so that the underlying manifold G is positively invariant under the stochastic system. Then, we formally derive the Fokker-Planck type equation defined on $$G\times \mathfrak {g}$$ in which $$\mathfrak {g}$$ denotes the Lie algebra corresponding to G. After identifying the target Fokker-Planck equation, we especially consider the unitary group $$G=\mathbf {U}(d)$$ and show that the equation on $$\mathbf {U}(d)$$ admits a global unique solution and that it can be rigorously derived using a stochastic mean-field limit procedure with a convergence rate of order $$O(1/\sqrt{N})$$. Finally, we restrict our concern to $$G=\mathbf {SU}(2)$$ to provide explicit calculation and present the nonlinear stability of an incoherent state for the Fokker-Planck equation depending on the relation between parameters.

A novel cell-centered finite volume scheme with positivity-preserving property for the anisotropic diffusion problems on general polyhedral meshes

In this paper, we propose a novel cell-centered positivity-preserving finite volume scheme for the anisotropic diffusion problems. The discretization of diffusion flux is based on the standard nonlinear two-point flux approximation. The cell vertexes are employed to define auxiliary unknowns. A positivity-preserving vertex interpolation algorithm is constructed to get the value of auxiliary unknowns. This interpolation algorithm has almost second order convergence rate on the distorted meshes. Numerical results illustrate that our scheme is efficient and practical in solving the 3D anisotropic diffusion problems.

A conservative compact finite difference scheme for the coupled Schrödinger-KdV equations

In this paper, a conservative compact finite difference scheme is presented to numerically solve the coupled Schrodinger-KdV equations. The analytic solutions of the coupled equations have some invariants such as the number of plasmons, the number of particles, and the energy of oscillations, and we proved that the compact difference scheme preserves those invariants in discrete sense. Optimal order convergence rate of the proposed linearized compact scheme was analyzed. Numerical experiments on model problems show that the scheme is of high accuracy.

Constructing higher order discontinuity-capturing schemes with upwind-biased interpolations and boundary variation diminishing algorithm

Based on the fifth-order scheme in our previous work (Deng et. al (2019) [28]), a new framework of constructing very high order discontinuity-capturing schemes is proposed for finite volume method. These schemes, so-called PnTm−BVD (polynomial of n-degree and THINC function of m-level reconstruction based on BVD algorithm), are designed by employing high-order upwind-biased interpolations and THINC (Tangent of Hyperbola for INterface Capturing) functions with adaptive steepness as the reconstruction candidates. The final reconstruction function in each cell is determined with a multi-stage BVD (Boundary Variation Diminishing) algorithm so as to effectively control numerical oscillation and dissipation. We devise the new schemes up to eleventh order in an efficient way by directly increasing the order of the underlying upwind scheme using high order polynomials. The analysis of the spectral property and accuracy tests show that the new reconstruction strategy well preserves the low-dissipation property of the underlying upwind schemes with high-order polynomials for smooth solution over all wave numbers and realizes n+1 order convergence rate. The performance of new schemes is examined through widely used benchmark tests, which demonstrate that the proposed schemes are capable of simultaneously resolving small-scale flow features with high resolution and capturing discontinuities with low dissipation. With outperforming results and simplicity in algorithm, the new reconstruction strategy shows great potential as an alternative numerical framework for computing nonlinear hyperbolic conservation laws that have discontinuous and smooth solutions of different scales.

Randomized exponential integrators for modulated nonlinear Schrödinger equations

Abstract We consider the nonlinear Schrödinger equation with dispersion modulated by a (formal) derivative of a time-dependent function with fractional Sobolev regularity of class $W^{\alpha ,2}$ for some $\alpha \in (0,1)$. Due to the loss of smoothness in the problem, classical numerical methods face severe order reduction. In this work, we develop and analyze a new randomized exponential integrator based on a stratified Monte Carlo approximation. The new discretization technique averages the high oscillations in the solution allowing for improved convergence rates of order $\alpha +1/2$. In addition, the new approach allows us to treat a far more general class of modulations than the available literature. Numerical results underline our theoretical findings and show the favorable error behavior of our new scheme compared to classical methods.

Higher order convergence rates in theory of homogenization II: Oscillatory initial data

We establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic Cauchy problems accompanied with rapidly oscillating initial data. Such result is new even for linear problems. Here we construct higher order initial layer and interior correctors, which describe the oscillatory behavior near the initial and interior time zone of the domain. To construct higher order correctors, we develop a regularity theory in macroscopic scales, and prove an exponential decay estimate for initial layer correctors.The higher order expansion requires an iteration process: successively correcting the initial layer, then the interior. This leads to a more complicated asymptotic expansion, as compared to the non-oscillating data case, and this complexity is present even in the linear case.A notable observation for fully nonlinear operators is that even if the given operator is space-time periodic, the interior correctors become aperiodic in the time variable as we proceed with the iteration process. Moreover, each interior corrector of higher order is paired with a space-time periodic version, and the difference between the two decays exponentially fast with time.

The energy-preserving finite difference methods and their analyses for system of nonlinear wave equations in two dimensions

The coupled sine-Gordon (SG) equations and the coupled Klein-Gordon (KG) equations play an important role in scientific fields, such as nonlinear optics, solid state physics, quantum mechanics. As their energies are conservative, it is of importance to develop energy preserving finite difference method (EP-FDM) for these systems of nonlinear wave equations. However, the energy preserving finite difference methods (EP-FDMs) for one-dimensional single sine-Gordon equation and one-dimensional single Klein-Gordon equation, can not directly be generalized to solve the systems of coupled SG or coupled KG equations, and the theoretical analysis technique used for 1D single SG equation or for 1D single KG equation is not suitable for the analysis of high dimensional problems. In this paper, we develop and analyze two kinds of energy preserving FDMs for the systems of coupled SG equations or coupled KG equations in two dimensions. One proposed scheme is a two-level scheme and the other is a three-level scheme. We prove the schemes to satisfy the energy conservations in the discrete forms. By using the fixed point theorem, it is shown that they are solvable. Also, it is further proved that they have the second order convergence rate in both time and space steps. Numerical tests show the performance of the methods and confirm the theoretical findings.

Convergence Analysis for Continuous-Time Markov Chain Approximation of Stochastic Local Volatility Models: Option Pricing and Greeks

This paper establishes the precise second order convergence rates of the continuous-time Markov chain (CTMC) approximation method for pricing options and calculating its Greeks under the general framework of stochastic local volatility models, which include the Heston and SABR models as special cases. Numerical examples confirm the theoretical findings.

A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model

<p style='text-indent:20px;'>In this work, a multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) is proposed to solve a coupled chemotaxis-fluid model. In the evolution equation of the proposed LBM, Beam-Warming (B-W) scheme is used to enhance the numerical stability. In numerical experiments, at first, the comparison between the classical LBM and the present LBM with B-W scheme is carried out by simulating blow up phenomenon of the Keller-Segel (K-S) model. Numerical results show that the stability of the present LBM with B-W scheme is better than the classical one. Then, the second order convergence rate of the proposed B-W scheme is verified in the numerical study of the coupled Navier-Stokes (N-S) K-S model. Finally, through solving the coupled chemotaxis-fluid model, the formation of falling bacterial plumes is numerically investigated. Numerical results agree well with existing ones in the literature.</p>

Multiscale mortar expanded mixed discretization of nonlinear elliptic problems

We consider the discretization of nonlinear second order elliptic partial differential equations by multiscale mortar expanded mixed method. This is a domain decomposition method in which the model problem is restricted to the small pieces by dividing the computational domain into the non-overlapping subdomains. An unknown (Lagrange multiplier) is introduced on the interfaces which serve as pressure Dirichlet boundary condition for local subdomain problems. A finite element space is defined on interface to approximate the pressure boundary such that the normal fluxes match weakly on interface.We demonstrated solvability of discretization, and established a priori L2-error estimates for both vector and scalar approximations. We proved the optimal order convergence rates by an appropriate choice of mortar space and polynomial degree of approximation. The uniqueness of discrete problem is shown for sufficiently small values of mesh size. An error estimate for the mortar pressure is derived via linear interface formulation with pressure dependent coefficient. We also present the analysis of the linear second order elliptic problem and prove the similar results. The computational experiments are presented to validate theory.

Tamed Runge-Kutta methods for SDEs with super-linearly growing drift and diffusion coefficients

Traditional explicit schemes such as the Euler-Maruyama, Milstein and stochastic Runge-Kutta methods, in general, result in strong and weak divergence when solving stochastic differential equations (SDEs) with super-linearly growing coefficients. Motivated by this, various modified versions of explicit Euler and Milstein methods were constructed and analyzed in the literature. In the present paper, we aim to introduce a family of explicit tamed stochastic Runge-Kutta (TSRK) methods for commutative SDEs with super-linearly growing drift and diffusion coefficients. Strong convergence rates of order 1.0 are successfully identified for the proposed methods under certain non-globally Lipschitz conditions. Compared to the Milstein-type methods involved with derivatives of coefficients, the newly proposed derivative-free TSRK methods can be computationally more efficient. Numerical experiments are reported to confirm the expected strong convergence rate of the TSRK methods.

Quasi-boundary value methods for regularizing the backward parabolic equation under the optimal control framework**Jun Liu’s research was supported by a Seed Grants for the Transitional and Exploratory Projects (STEP) Award (FY2019) from the SIUE Graduate School. Mingqing Xiao’s research was supported in part by the NSF-DMS140928, 1854638 of the United States.

In this paper, we propose to solve the classical backward parabolic equation under the optimal control framework associated with the Tikhonov regularization formulation, where the initial condition is set to be the control variable while the misfit of the final condition is formulated as the Tikhonov objective functional to be minimized. The corresponding first-order necessary optimality system is discretized in a one-shot manner by a second-order finite difference scheme in space and time. The proposed optimal control setting, as discussed in the paper, provides more general framework in terms of solving this type of ill-posed inverse problem. In particular, several existing nonlocal quasi-boundary value methods appear to be the special cases of our proposed optimal control framework with appropriately chosen generalized regularization setting respectively. The relationship between our Tikhonov regularization approach and the quasi-boundary value methods is discussed in detail. The optimal control approach based on Tikhonov regularization is shown to deliver the known optimal order convergence rate. Numerical examples are provided to demonstrate both effective accuracy of approximation as well as excellent stability under the optimal control realization, comparing with three recent quasi-boundary value methods in literature.

Quasi-DNS Dataset of a Piloted Flame with Inhomogeneous Inlet Conditions

A quasi-DNS of the partially premixed turbulent Sydney flame in configuration FJ200-5GP-Lr75-57 has been conducted using detailed molecular diffusion for multi-component mixtures and complex reaction mechanisms. In order to study flame dynamics like regime transition in this flame for the development of new combustion models and to directly compare the quasi-DNS to different LES models, the simulation results are compiled into a data base. Because the simulation was performed with OpenFOAM, we demonstrate the quasi-DNS capabilities of OpenFOAM by performing canonical test cases. They attest that OpenFOAM’s cubic discretization has lower numerical diffusion compared to classical central difference schemes and can reach higher than second order convergence rate in some cases. The quasi-DNS of the Sydney flame is conducted with a self-developed reacting flow solver which is able to accurately compute molecular diffusion coefficients from kinetic gas theory and employs a fast implementation for detailed reaction mechanisms. The computational mesh is shown to be able to resolve the flow as well as the flame front sufficiently for the quasi-DNS. Comparisons with experimental data also show that the simulation can quantitatively reproduce measured time-mean and time-RMS statistics.

A Priori Error Estimates for an Optimal Control Problem Governed by a Variational Inequality of the Second Kind

AbstractWe consider an optimal control problem governed by an elliptic variational inequality of the second kind. The problem is discretized by linear finite elements for the state and a variational discrete approach for the control. We derive nearly optimal a priori error estimates based on L∞‐error estimates for the variational inequality and a quadratic growth condition. Our error analysis yields a convergence rate of order 1 − ϵ for the L2‐norm of the control.