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This paper focuses on numerical algorithms to value American options under regime switching. The prices of such options satisfy a set of complementary parabolic problems on an unbounded domain. Based on our previous experience, the pricing model could be truncated into a linear complementarity problem (LCP) over a bounded domain. In addition, we transform the resulting LCP into an equivalent variational problem (VP), and discretize the VP by an Euler‐finite element method. Since the variational matrix in the discretized system is P‐matrix, a primal‐dual active set (PDAS) algorithm is proposed to evaluate the option prices efficiently. As a specialty of PDAS, the optimal exercise boundaries in all regimes are obtained without further computation cost. Finally, numerical simulations are carried out to test the performance of our proposed algorithm and compare it to existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fast Alikhanov algorithm with nonuniform time steps for a two dimensional distributed‐order time–space fractional advection–dispersion equation. First, an efficient fast Alikhanov algorithm on the general nonuniform time steps for the evaluation of Caputo fractional derivative is presented to sharply reduce the computational work and storage, and are applied to the distributed‐order time fractional derivative or multi‐term time fractional derivative under the nonsmooth regularity assumptions. And a generalized discrete fractional Grönwall inequality is extended to multi‐term fractional derivative or distributed‐order fractional derivative for analyzing theoretically our algorithm. Then the stability and convergence of time semi‐discrete scheme are investigated. Furthermore, we derive the corresponding fully discrete scheme by finite element method and discuss its convergence. At last, the given numerical examples adequately confirm the correctness of theoretical analysis and compare the computing effectiveness between the fast algorithm and the direct method. [ABSTRACT FROM AUTHOR]

An efficient spectral method is developed in this paper for the two‐dimensional Stokes eigenvalues on arbitrary triangle. By using the spectral theory of compact operator and approximate property of orthogonal polynomial, we give the error estimate of the approximate eigenvalues and eigenfunctions. In addition, we also present some numerical results to show the validity of our algorithm and the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

Comparing with the classic phase filed models, the fractional models such as time fractional Allen–Cahn and Cahn–Hilliard equations are equipped with Caputo fractional derivative and can describe more practical phenomena for modeling phase transitions. In this paper, we construct two accurate and efficient linear algorithms for the time fractional Cahn–Hilliard and Allen–Cahn equations with general nonlinear bulk potential. The main contribution is that we have proved the unconditional energy stability for the time fractional Cahn–Hilliard and Allen–Cahn models and their semi‐discrete schemes carefully and rigorously. Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of our proposed schemes. [ABSTRACT FROM AUTHOR]

This paper focuses on the numerical solution of the variable coefficient multiterm time fractional advection–diffusion equation via exponential B‐splines. We discretize the temporal part by using the Crank–Nicolson method and spatial part by the exponential B‐splines. The unconditional stability is obtained by the Von‐Neumann method. The convergence rates are also studied. Numerical simulations confirm the theoretically expected accuracy in both time and space directions. A comparative analysis with the other methods shows the superiority of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical scheme which is an improved minimax method proposed by Yao‐Zhou's study along with Newton's method for finding the multiple solutions of p‐area problems. It is combined with Newton's method since the local minimax method stagnates close to a critical point of the functional whereas Newton's method converges rapidly close to a critical point. We discuss the theoretical convergence result of this proposed hybrid scheme and the convergence of the finite element based solutions as well. Further, we carry out instability analysis of the unstable solutions through their local minimax type characterizations, where the instability behavior of a solution can be analyzed before calculating the solution. This analysis also provides ways to numerically compute the Morse index of the solutions. Finally, numerical experiments are performed to demonstrate the algorithm and the theoretical results. [ABSTRACT FROM AUTHOR]

Solutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ⊂ ℝ3 with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H1(Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions. [ABSTRACT FROM AUTHOR]

This paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper we try to redesign an algorithm analyzed in previous works by Amat et al. in order to obtain a centered strategy aimed to approximate the solution of linear and nonlinear systems of conservation laws. We perform the analysis of the new algorithm in order to explain why the new strategy works close to shocks in the solution of conservation laws. The objective is to force the use of the most centered stencil of those used whenever the data are affected by the presence of a high gradient. [ABSTRACT FROM AUTHOR]

In this paper, we propose an effective spectral method based on dimension reduction scheme for fourth order problems in polar geometric domains. First, the original problem is decomposed into a series of one‐dimensional fourth order problems by polar coordinate transformation and the orthogonal properties of Fourier basis function. Then the weak form and the corresponding discrete scheme of each one‐dimensional fourth order problem are derived by introducing polar conditions and appropriate weighted Sobolev spaces. In addition, we define the projection operators in the weighted Sobolev space and give its approximation properties, and further prove the error estimation of each one‐dimensional fourth order problem. Finally, we provide some numerical examples, and the numerical results show the effectiveness of our algorithm and the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is proposed for the error estimates of the element‐free Galerkin method for a quasistatic contact problem with the Tresca friction. The penalty method is used to impose the clamped boundary conditions. The duality algorithm is also given to deal with the non‐differentiable term in the quasistatic contact problem with the Tresca friction. The error estimates indicate that the convergence order is dependent on the nodal spacing, the time step, the largest degree of basis functions in the moving least‐squares approximation, and the penalty factor. Numerical examples demonstrate the effectiveness of the element‐free Galerkin method and verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider the effect of adding a coarse mesh correction to the two‐grid algorithm for the mixed Navier–Stokes/Darcy model. The method yields both L2 and H1 optimal velocity and piezometric head approximations and an L2 optimal pressure approximation. The method involves solving one small, coupled, nonlinear coarse mesh problem, two independent subproblems (linear Navier–Stokes equation and Darcy equation) on the fine mesh, and a correction problem on the coarse mesh. Theoretical analysis and numerical tests are done to indicate the significance of this method. [ABSTRACT FROM AUTHOR]

An enhanced finite‐difference time‐domain (FDTD) algorithm is built to solve the transverse electric two‐dimensional Maxwell's equations with inhomogeneous dielectric media where the electric fields are discontinuous across the dielectric interface. The new algorithm is derived based upon the integral version of the Maxwell's equations as well as the relationship between the electric fields across the interface. To resolve the instability issue of Yee's scheme (staircasing) caused by discontinuous permittivity across the interface, our algorithm revises the permittivities and makes some corrections to the scheme for the cells around the interface. It is also an improvement over the contour‐path effective permittivity algorithm by including some extra terms in the formulas. The scheme is validated in solving the scattering of a dielectric cylinder with exact solution from Mie theory and is then compared with the above contour‐path method, the usual staircasing and the volume‐average method. The numerical results demonstrate that the new algorithm has achieved significant improvement in accuracy over other methods. Furthermore, the algorithm has a simple structure and can be merged into current FDTD software packages easily. The C++ source code for this paper is provided as supporting information for public access. [ABSTRACT FROM AUTHOR]

In this paper we introduce fast numerical algorithms for the solution of the model. For each variable, background illumination, amplitude modulation and phase map, we develop a fixed point method. Then, we write all three algorithms in the same framework and analyze their convergence rates, local smoothing factors by means of local Fourier analysis and present experimental evidence of their performance on synthetic and real world problems. [ABSTRACT FROM AUTHOR]

In this paper, high‐order numerical methods for time‐Caputo and space‐Riesz fractional Bloch‐Torrey equations in one‐ and two‐dimensional space are constructed, where the second‐order backward fractional difference operator and the sixth‐order fractional‐compact difference operator are applied to approximate the time and space fractional derivatives, respectively. The stability and convergence of the methods are analyzed and it is shown that the convergence orders are higher than the earlier work. Finally, some numerical experiments are presented to demonstrate the effectiveness of the methods and confirm our theoretical results. [ABSTRACT FROM AUTHOR]

To investigate the fluid–fluid interaction problem, we study two first‐order in time, decoupled algorithms for a coupling system. This system consist of two heat equations, which are coupled by a condition that allows energy to pass back and forth across the interface. The first scheme is a combination of the three‐level implicit method with the coupling terms treated by the explicit leap‐frog method. Under a time step size condition, we prove that it is stable and first‐order convergent. To remove the time step size condition, we modify the treatment of the coupling interface term and propose a second scheme. Instead of making the interface term explicitly, we only lag the variable in the different subdomain onto the previous time step. As expected, the second scheme is proved to be stable and convergent unconditionally. Since both schemes only require the solution of two decoupled heat equations at each time step. Hence, they are efficient in computation and can be easily implemented by using legacy codes. The numerical tests also confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we design, analyze, and test the multiwavelets Galerkin method to solve the two‐dimensional Burgers equation. Using Crank–Nicolson scheme, time is discretized and a PDE is obtained for each time step. We use the multiwavelets Galerkin method for solving these PDEs. Multiwavelets Galerkin method reduces these PDEs to sparse systems of algebraic equations. The cost of this method is proportional to the number of nonzero coefficients at each time step. The results illustrate, by selecting the appropriate threshold while the number of nonzero coefficients reduces, the error will not be less than a certain amount. The L2 stability and convergence of the scheme have been investigated by the energy method. Illustrative examples are provided to verify the efficiency and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, a weak Galerkin (WG) method is first presented and analyzed for the quasi‐linear elliptic problem of non‐monotone type. By using Brouwer's fixed point technique, the existence of WG solution and error estimates in both the energy norm and the L2$$ {L}^2 $$ norm are derived. Then an efficient two‐grid WG method is introduced to improve the computational efficiency. The convergence error of the two‐grid WG method is analyzed in the energy norm. Numerical experiments are presented to verify our theoretical findings. [ABSTRACT FROM AUTHOR]

Fractional equations have become the model of choice in several applications where heterogeneities at the microstructure result in anomalous diffusive behavior at the macroscale. In this work we introduce a new fractional operator characterized by a doubly‐variable fractional order and possibly truncated interactions. Under certain conditions on the model parameters and on the regularity of the fractional order we show that the corresponding Poisson problem is well‐posed. We also introduce a finite element discretization and describe an efficient implementation of the finite‐element matrix assembly in the case of piecewise constant fractional order. Through several numerical tests, we illustrate the improved descriptive power of this new operator across media interfaces. Furthermore, we present one‐dimensional and two‐dimensional h‐convergence results that show that the variable‐order model has the same convergence behavior as the constant‐order model. [ABSTRACT FROM AUTHOR]

In this article, three novel algorithms are developed and successfully applied to investigate the stable solutions of time‐fractional nonlinear telegraph equations. Firstly, we presented the shifted Gegenbauer polynomials through appropriate transformations. The approximation of a function u(||x||, t) is defined via shifted Gegenbauer polynomials (SGPs) and then developed the operational matrices of a positive integer and non‐integer derivatives assisted by multi‐dimensional shifted Gegenbauer polynomials vectors. Linearized fully spectral method (LFSM) is proposed by means of function approximations and developed operational matrices whilst Picard iterative scheme is used to linearize the discussed problem. To introduce a semi‐discrete method (SDM), the temporal derivative is approximated through a central difference scheme whereas spatial derivatives are approximated assisted by novel operational matrices. A fully numerical method (FNM) is proposed by introducing the central difference for temporal and spatial variables. Proposed computational methods transform the time‐fractional nonlinear telegraph equation into a system of linear algebraic equations that are easier to tackle. An inclusive comparative study is witnesses that suggested schemes are effective, accurate and well‐matched to investigate the approximate solutions of the problems udnerstudy. Convergence and error bound of the suggested methods are investigated theoretically whereas stability analysis is performed numerically. Moreover, the developed methods can be used conveniently to examine the numerical solution of other multi‐dimensional highly nonlinear fractional or variable order problems of physical nature. [ABSTRACT FROM AUTHOR]

The discrete variational inequalities resulting from C0 interior penalty methods for the obstacle problem of clamped Kirchhoff plates can be solved by the primal‐dual active set algorithm. We develop and analyze additive Schwarz preconditioners for the auxiliary systems that appear in each iteration of the primal‐dual active set algorithm. Numerical results corroborate our theoretical estimates. [ABSTRACT FROM AUTHOR]

In this article, barycentric rational interpolation and local radial basis functions (RBFs) based numerical algorithms are developed for solving multidimensional sine‐Gordon (SG) equation. In the development of these algorithms, the first step is to drive a semi‐discretization in time with a finite difference, and then the semi‐discrete problem is analyzed for truncation errors and convergence in L2 and H1 spaces. After that, the semi‐discrete system is fully discretized by two different functions, such as barycentric rational and local RBFs. Finally, we obtain a linear system in both the algorithms and the system is solved by a MATLAB routine. In numerical experiments, 1D and 2D SG are considered with various examples of line and ring solitons. Moreover, a comparative study of present results with available numerical ones and exact solutions is also discussed. [ABSTRACT FROM AUTHOR]

In this work we develop an algorithm to compute analytical solution for a two‐dimensional fuzzy heat equation involving external source term with fractional order. The algorithm is based on the Homotopy perturbation method: The required result is computed in series form which is rapidly convergent to the exact solution. Examples are given to verify the result, which are compared with the exact solution to illustrate the efficiency and the capability of the proposed method. [ABSTRACT FROM AUTHOR]

In this work, numerical solution of nonlinear modified Burgers equation is obtained using an improvised collocation technique with cubic B‐spline as basis functions. In this technique, cubic B‐splines are forced to satisfy the interpolatory condition along with some specific end conditions. Crank–Nicolson scheme is used for temporal domain and improvised cubic B‐spline collocation method is used for spatial domain discretization. Quasilinearization process is followed to tackle the nonlinear term in the equation. Convergence of the technique is established to be of order O(h4 + Δt2). Stability of the technique is examined using von‐Neumann analysis. L2 and L∞ error norms are calculated and are compared with those available in existing works. Results are found to be better and the technique is computationally efficient, which is shown by calculating CPU time. [ABSTRACT FROM AUTHOR]

In this article, we develop a Crank–Nicolson alternating direction implicit finite volume method for time‐dependent Riesz space‐fractional diffusion equation in two space dimensions. Norm‐based stability and convergence analysis are given to show that the developed method is unconditionally stable and of second‐order accuracy both in space and time. Furthermore, we develop a lossless matrix‐free fast conjugate gradient method for the implementation of the numerical scheme, which only has O(N) memory requirement and O(NlogN) computational complexity per iteration with N being the total number of spatial unknowns. Several numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed scheme for large‐scale modeling and simulations. [ABSTRACT FROM AUTHOR]

We have developed a new numerical method based on Haar wavelet (HW) in this article for the numerical solution (NS) of one‐ and two‐dimensional hyperbolic Telegraph equations (HTEs). The proposed technique is utilized for one‐ and two‐dimensional linear and nonlinear problems, which shows its advantage over other existing numerical methods. In this technique, we approximated both space and temporal derivatives by the truncated Haar series. The algorithm of the method is simple and we can implement easily in any other programming language. The technique is tested on some linear and nonlinear examples from literature. The maximum absolute errors (MAEs), root mean square errors (RMSEs), and computational convergence rate are calculated for different number of collocation points (CPs) and also some 3D graphs are also drawn. The results show that the proposed technique is simply applicable and accurate. [ABSTRACT FROM AUTHOR]

In this article, a spatial two‐grid finite element (TGFE) algorithm is used to solve a two‐dimensional nonlinear space–time fractional diffusion model and improve the computational efficiency. First, the second‐order backward difference scheme is used to formulate the time approximation, where the time‐fractional derivative is approximated by the weighted and shifted Grünwald difference operator. In order to reduce the computation time of the standard FE method, a TGFE algorithm is developed. The specific algorithm is to iteratively solve a nonlinear system on the coarse grid and then to solve a linear system on the fine grid. We prove the scheme stability of the TGFE algorithm and derive a priori error estimate with the convergence result O(Δt2 + hr + 1 − η + H2r + 2 − 2η). Finally, through a two‐dimensional numerical calculation, we improve the computational efficiency and reduce the computation time by the TGFE algorithm. [ABSTRACT FROM AUTHOR]

Orthogonal spline collocation is implemented for the numerical solution of two‐dimensional Helmholtz problems with discontinuous coefficients in the unit square. A matrix decomposition algorithm is used to solve the collocation matrix system at a cost of O(N2 log N) on an N × N partition of the unit square. The results of numerical experiments demonstrate the efficacy of this approach, exhibiting optimal global estimates in various norms and superconvergence phenomena for a broad spectrum of wave numbers. [ABSTRACT FROM AUTHOR]

An initial boundary value problem is considered for the dynamic beam system wtt=cd−a+b∫01wx2dxwxx−cdψx,ψtt=cψxx−c2d(ψ−wx). Its solution is found by means of an algorithm, the constituent parts of which are the finite element method, the implicit symmetric difference scheme used to approximate the solution with respect to the spatial and time variables, and also a Picard type iteration process for solving the system of nonlinear equations obtained by discretization. Errors of three parts of the algorithm are estimated and, as a result, its total error estimate is obtained. A numerical example is solved. [ABSTRACT FROM AUTHOR]

In this work, we propose a hybrid radial basis functions (RBFs) collocation technique for the numerical solution of fractional advection–diffusion models. In the formulation of hybrid RBFs (HRBFs), there exist shape parameter (c*) and weight parameter (ϵ) that control numerical accuracy and stability. For these parameters, an adaptive algorithm is developed and validated. The proposed HRBFs method is tested for numerical solutions of some fractional Black–Sholes and diffusion models. Numerical simulations performed for several benchmark problems verified the proposed method accuracy and efficiency. The quantitative analysis is made in terms of L∞, L2, Lrms, and Lrel error norms as well as number of nodes N over space domain and time‐step δt. Numerical convergence in space and time is also studied for the proposed method. The unconditional stability of the proposed HRBFs scheme is obtained using the von Neumann methodology. It is observed that the HRBFs method circumvented the ill‐conditioning problem greatly, a major issue in the Kansa method. [ABSTRACT FROM AUTHOR]

New algorithms, combining asymptotic numerical method (ANM) and method of fundamental solutions, are proposed to compute bifurcation points on branch solutions of a nonlinear bi‐harmonic problem. Three methods, mainly based on asymptotic developments framework, are then proposed. The first one consists in exploiting the ANM step accumulation close to the bifurcation points on a solution branch, the second method allows the introduction of an indicator that vanishes at the bifurcation points, and finally the first real root of the Padé approximant denominator represents the third bifurcation indicator. Two numerical examples are considered to analyze the robustness of these algorithms. [ABSTRACT FROM AUTHOR]

Multiscale or multiphysics problems often involve coupling of partial differential equations posed on domains of different dimensionality. In this work, we consider a simplified model problem of a 3d‐1d coupling and the main objective is to construct algorithms that may utilize standard multilevel algorithms for the 3d domain, which has the dominating computational complexity. Preconditioning for a system of two elliptic problems posed, respectively, in a three‐dimensional domain and an embedded one dimensional curve and coupled by the trace constraint is discussed. Investigating numerically the properties of the well‐defined discrete trace operator, it is found that negative fractional Sobolev norms are suitable preconditioners for the Schur complement of the system. The norms are employed to construct a robust block diagonal preconditioner for the coupled problem. [ABSTRACT FROM AUTHOR]

Recently, many new applications in engineering and science are governed by a series of time‐fractional partial integrodifferential equations, which will lead to new challenges for numerical simulation. In this article, we propose and analyze an efficient iterative algorithm for the numerical solutions of such equations subject to initial and Dirichlet boundary conditions. The algorithm provide appropriate representation of the solutions in infinite series formula with accurately computable structures. By interrupting the n ‐term of exact solutions, numerical solutions of linear and nonlinear time‐fractional equations of nonhomogeneous function type are studied from mathematical viewpoint. Convergence analysis, error estimations, and error bounds under some hypotheses which provide the theoretical basis of the proposed algorithm are also discussed. The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such integrodifferential equations. [ABSTRACT FROM AUTHOR]

In this article, a fast‐iterative method and a fast‐direct method is proposed for solving one‐dimensional and two‐dimensional tempered fractional diffusion equations with constant coefficients. The proposed iterative method is accelerated by circulant preconditioning which is shown to converge superlinearly while the proposed direct method is based on circulant and skew‐circulant representation for Toeplitz matrix inversion. In one‐dimensional case, the operation cost of the proposed methods are both shown to be O ( N log N ) with O ( N ) memory requirement in each time step, where N is the number of spatial nodes. With the alternating direction implicit method, it is proven that the proposed fast solution algorithms can be extended to handle two‐dimensional tempered fractional diffusion equations with O ( N 2 log N ) operation cost and O ( N 2 ) memory requirement in each time step, where the number of spatial nodes in x ‐direction and y ‐direction both equal to N . Numerical examples are provided to illustrate the effectiveness and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

The multisymplectic schemes have been used in numerical simulations for the RLW-type equation successfully. They well preserve the local geometric property, but not other local conservation laws. In this article, we propose three novel efficient local structure-preserving schemes for the RLW-type equation, which preserve the local energy exactly on any time-space region and can produce richer information of the original problem. The schemes will be mass- and energy-preserving as the equation is imposed on appropriate boundary conditions. Numerical experiments are presented to verify the efficiency and invariant-preserving property of the schemes. Comparisons with the existing nonconservative schemes are made to show the behavior of the energy affects the behavior of the solution.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1678-1691, 2017 [ABSTRACT FROM AUTHOR]

This article is concerned with the convergence of the level-set algorithm introduced by Aslam (J Comput Phys 167 (2001), 413-438) for tracking the discontinuities in scalar conservation laws in the case of linear or strictly convex flux function. The numerical method is deduced by the level-set representation of the entropy solution: the zero of a level-set function is used as an indicator of the discontinuity curves and two auxiliary states, which are assumed continuous through the discontinuities, are introduced. We rewrite the numerical level-set algorithm as a procedure consisting of three big steps: (a) initialization, (b) evolution, and (c) reconstruction. In (a), we choose an entropy admissible level-set representation of the initial condition. In (b), for each iteration step, we solve an uncoupled system of three equations and select the entropy admissible level-set representation of the solution profile at the end of the time iteration. In (c), we reconstruct the entropy solution using the level-set representation. We prove the convergence of the numerical solution to the entropy solution in [ABSTRACT FROM AUTHOR]