Backward Euler Method Research Articles

Novel numerical techniques based on mimetic finite difference method for pricing two dimensional options

The Black–Scholes differential operator which underlies the option pricing of European and American options is known to be degenerate close to the boundary at zero. At this singularity, important properties of the differential operator are lost and the classical finite difference scheme applied to this problem fails to give accurate approximations as it is no longer monotone. In this paper novel numerical techniques based on mimetic finite difference method are proposed for accurately pricing European and American options. More precisely, we propose the mimetic and fitted mimetic finite difference methods, which are techniques that preserve and conserve general properties of the continuum operator in the discrete case. The fitted method further handles the degeneracy of the underlying partial differential equations (PDE). Those spatial discretization methods are coupled with the Euler implicit method for time discretization. Several numerical simulations are performed to demonstrate the robustness of our methods comparing to standard fitted finite volume method for both European and American put options.

Parameter-Uniform Numerical Scheme for Singularly Perturbed Delay Parabolic Reaction Diffusion Equations with Integral Boundary Condition

Numerical computation for the class of singularly perturbed delay parabolic reaction diffusion equations with integral boundary condition has been considered. A parameter-uniform numerical method is constructed via the nonstandard finite difference method for the spatial direction, and the backward Euler method for the resulting system of initial value problems in temporal direction is used. The integral boundary condition is treated using numerical integration techniques. Maximum absolute errors and the rate of convergence for different values of perturbation parameter ε and mesh sizes are tabulated for two model examples. The proposed method is shown to be parameter-uniformly convergent.

Fast iterative solver for the optimal control of time‐dependent PDEs with Crank–Nicolson discretization in time

AbstractIn this article, we derive a new, fast, and robust preconditioned iterative solution strategy for the all‐at‐once solution of optimal control problems with time‐dependent PDEs as constraints, including the heat equation and the non‐steady convection–diffusion equation. After applying an optimize‐then‐discretize approach, one is faced with continuous first‐order optimality conditions consisting of a coupled system of PDEs. As opposed to most work in preconditioning the resulting discretized systems, where a (first‐order accurate) backward Euler method is used for the discretization of the time derivative, we employ a (second‐order accurate) Crank–Nicolson method in time. We apply a carefully tailored invertible transformation for symmetrizing the matrix, and then derive an optimal preconditioner for the saddle‐point system obtained. The key components of this preconditioner are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to apply this latter approximation—these are constructed using our work in transforming the matrix system. We prove the optimality of the approximation of the Schur complement through bounds on the eigenvalues, and test our solver against a widely‐used preconditioner for the linear system arising from a backward Euler discretization. These demonstrate the effectiveness and robustness of our solver with respect to mesh‐sizes, regularization parameter, and diffusion coefficient.

Parameter‐robust preconditioning for unsteady Stokes control problems

AbstractWe propose a saddle‐point preconditioner for an optimization problem constrained by the time‐dependent Stokes equations, discretized using the backward Euler method in time. The key ingredients are an inner iteration for the (1, 1)‐block, accelerated by a known preconditioner for the heat control problem, and an approximation of the Schur complement involving a commutator argument applied to a block matrix. Numerical results demonstrate the efficacy and robustness of this approach.

An efficient Sinc‐collocation method by the single exponential transformation for the nonlinear fourth‐order partial integro‐differential equation with multiterm kernels

In this article, a Sinc‐collocation method is proposed and analyzed for solving the nonlinear fourth‐order partial integro‐differential equation with the multiterm kernels. The time derivative is discretized by the backward Euler method, and the Riemann–Liouville (R‐L) fractional integral terms are treated by means of the first‐order convolution quadrature. Then, the nonlinear term is linearized. A fully discrete scheme is established with the space discretization by the Sinc approximation based on the single exponential (SE) transformation. The spatial exponential convergence of the proposed method is derived. Several numerical examples are provided to illustrate the accuracy and effectiveness of our method.

A Fast Preconditioned Semi-Implicit Difference Scheme for Strongly Nonlinear Space-Fractional Diffusion Equations

In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from O(Ns2) to O(Ns) and the computational complexity from O(Ns3) to O(NslogNs) in each iterative step, where Ns is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method.

Numerical analysis of a fully discrete stabilized FEM for system of singularly perturbed parabolic IBVPs

In this article, a fully discrete numerical method is studied to solve the system of singularly perturbed parabolic convection–diffusion problem. The full discretization incorporates the backward-Euler method and the streamline-diffusion finite element method (SDFEM) for the temporal and spatial discretization, respectively. In each equation, the highest ordered spatial derivatives are multiplied by distinct parameters, hence the overlapping boundary layers phenomena displayed in the solution. To capture the singularity of the solution, a layer-adapted mesh has been used for the spatial domain discretization. The stability of the numerical method has been addressed by considering an appropriate stabilization parameter. A complete error analysis has shown the first order of accuracy of the fully discrete method in the discrete -norm. Lastly, few numerical experiments have been addressed to justify the theoretical estimates.

Refactorization of a variable step, unconditionally stable method of Dahlquist, Liniger and Nevanlinna

The one-leg, two-step time-stepping scheme proposed by Dahlquist, Liniger and Nevanlinna has clear advantages in complex, stiff numerical simulations: unconditional G-stability for variable time-steps and second-order accuracy. Yet it has been underutilized due, partially, to its complexity of direct implementation. We prove herein that this method is equivalent to the backward Euler method with pre- and post arithmetic steps added. This refactorization eases implementation in complex, possibly legacy codes. The realization we develop reduces complexity, including cognitive complexity and increases accuracy over both first order methods and constant time steps second order methods.

Research on real time simulation modeling method of large scale MMC electromagnetic transient

Modular multilevel converter (MMC) contains a large number of power electronic switching devices. The modeling method based on switching circuit model needs a lot of resources and the simulation speed is slow, so it is difficult to realize large-scale real-time simulation of electromagnetic transient. A MMC electromagnetic transient numerical modeling method based on ideal transformer model (ITM) is presented. Firstly, the MMC system is divided into the main circuit network and the sub module group network by ITM method, and the error caused by decoupling delay in serial and parallel real time simulation is compensated respectively by interpolation prediction and advanced interpolation prediction. Secondly, the capacitor in sub module is discreted respectively by trapezoidal integration method, backward Euler method and Gear-2 method. Based on the above numerical integration, the difference equations of capacitance voltage, capacitance current and output voltage of half bridge and full bridge sub modules are derived. Then, in order to improve the calculation speed, a simplified numerical model of half bridge and full bridge sub module based on switching function is proposed. Finally, the MMC based on switching circuit model runs off-line simulation in the simulation software, and the above MMC numerical modeling method runs real-time simulation in Speedgoat real-time simulator. The off-line and real-time simulation results of the MMC numerical modeling method and the switching circuit model are compared. And the simulation results verify the feasibility and effectiveness of the above MMC numerical modeling method in real time simulation.

Numerical Treatment on Parabolic Singularly Perturbed Differential Difference Equation via Fitted Operator Scheme

This paper proposes a new fitted operator strategy for solving singularly perturbed parabolic partial differential equation with delay on the spatial variable. We decomposed the problem into three piecewise equations. The delay term in the equation is expanded by Taylor series, the time variable is discretized by implicit Euler method, and the space variable is discretized by central difference methods. After developing the fitting operator method, we accelerate the order of convergence of the time direction using Richardson extrapolation scheme and obtained O h 2 + k 2 uniform order of convergence. Finally, three examples are given to illustrate the effectiveness of the method. The result shows the proposed method is more accurate than some of the methods that exist in the literature.

Backward Euler method for the equations of motion arising in Oldroyd model of order one with nonsmooth initial data

Abstract In this paper, a backward Euler method combined with finite element discretization in spatial direction is discussed for the equations of motion arising in the two-dimensional Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^{\infty }$ in time. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal a priori error estimate in $\textbf {L}^2$-norm is derived for the discrete problem with nonsmooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition. Finally, we present some numerical results to validate our theoretical results.

Anisotropic hyperelastic/plastic behavior on stress-constrained thin structures by iterating on the elastic Cauchy–Green tensor

Thin-walled structures impose zero-stress constraints at free faces. These are in general nonlinear functions of strain and, in the elasto-plastic case, non-smooth. Under these circumstances, the conjugate strain components become constitutive unknowns. By iterating for the elastic Cauchy–Green tensor in the 6−dimensional space, we directly obtain a system that includes the zero-stress constraints. Coupling of the stress and strain constitutive unknowns is possible due to the use of a convenient equation arrangement. This contrasts with the standard decoupling schemes. Herein, we use the Kröner–Lee decomposition of the deformation gradient to obtain a differential–algebraic system which includes stress constraints in analogous form to additional yield conditions. The source is the right Cauchy–Green tensor. The main contribution is the development of a thin-walled implicit integrator for any hyperelastic case and any yield function. Integration is based on a backward-Euler method for the flow law complemented by the solution of a yield condition and the stress constraints. We make use of the elastic Mandel stress construction, which is power-consistent with the plastic strain rate. Two complete fully orthotropic worked examples are presented.

Formulative Visualization of Numerical Methods for Solving Non-Linear Ordinary Differential Equations

Many physical problems in the real world are frequently modeled by ordinary differential equations (ODEs). Real-life problems are usually non-linear, numerical methods are therefore needed to approximate their solution. We consider different numerical methods viz., Explicit (Forward) and Implicit (Backward) Euler method, Classical second-order Runge-Kutta (RK2) method (Heun’s method or Improved Euler method), Third-order Runge-Kutta (RK3) method, Fourth-order Runge-Kutta (RK4) method, and Butcher fifth-order Runge-Kutta (BRK5) method which are popular classical iteration methods of approximating solutions of ODEs. Moreover, an intuitive explanation of those methods is also be presented, comparing among them and also with exact solutions with necessary visualizations. Finally, we analyze the error and accuracy of these methods with the help of suitable mathematical programming software.

On analysis of Persidskii systems and their implementations using LMIs

The conditions of (integral) input-to-state stability and input–output-to-state stability are established for a class of generalized Persidskii systems. The proposed conditions are formulated using linear matrix inequalities. Based on these results the conditions of convergence are derived for discretizations of this class of models obtained by the explicit and implicit Euler methods. The proposed theory is finally applied to design a robust stabilization control.

A comprehensive implicit substepping integration scheme for multisurface plasticity

AbstractA complex elastoplastic model requires a robust integration procedure of the evolution equations. The performance of the finite element solution is directly affected by the convergence characteristics of the state‐update procedure. Thereby, this study proposes a comprehensive numerical integration scheme to deal with generic multisurface plasticity models. This algorithm is based on the backward Euler method aiming at accuracy and stability, and on the Newton–Raphson method to solve the unconstrained optimization problem. In this scenario, a line search strategy is adopted to improve the convergence characteristics of the algorithm. The golden section method, an exact line search, is considered. Also, a substepping scheme is implemented to provide additional robustness to the state‐update procedure. Therefore, this work contributes to computational plasticity presenting an adaptive substep size scheme and a consistent tangent modulus according to the substepping technique. Finally, some numerical problems are evaluated using the proposed algorithm. Single‐surface and novel multisurface plasticity models are employed in these analyses. The results testify how the line search and substepping strategies can improve the robustness of the nonlinear analysis.

Collocation method using artificial viscosity for time dependent singularly perturbed differential–difference equations

A parameter uniform numerical method is presented for solving singularly perturbed time-dependent differential–difference equations with small shifts. To approximate the terms with the shifts, Taylor’s series expansion is used. The resulting singularly perturbed parabolic partial differential equation is solved using an implicit Euler method in temporal direction and cubic B-spline collocation method for the resulting system of ordinary differential equations in spatial direction, and an artificial viscosity is introduced in the scheme using the theory of singular perturbations. The proposed method is shown to be accurate of order OΔt+h2 by preserving ɛ-uniform convergence, where h and Δt denote spatial and temporal step sizes, respectively. Several test examples are solved to demonstrate the effectiveness of the proposed method. The computed numerical results show that the proposed method provides more accurate results than some methods exist in the literature and suitable for solving such problems with little computational effort.

Evaluation of an effective and robust implicit time-integration numerical scheme for Navier-Stokes equations in a CFD solver for compressible flows

The present work describes the implementation of an implicit time-integration numerical scheme to solve viscous flows in an in-house CFD solver. The scheme is developed to calculate engineering problems involving compressible flows. This work extends the defect-correction technique for the 3D flow calculations, and all mathematical formulations are described. The CFD solver is based on the finite-volume method (FVM) to calculate the three-dimensional flow and can be applied to solve unstructured meshes. The current implementation uses the Flux-Difference Splitting method (FDS) developed by Roe combined with the MUSCL method and the Venkatakrishnan flux limiters to provide better accuracy of the numerical solutions. The implicit time-integration scheme was linearized applying the backward Euler method on the left-hand side (LHS) and a Newton-type linearization on the right-hand side (RHS) of the governing equations. The Jacobian matrix was computed analytically for the inviscid fluxes using the Roe fluxes, and for the viscous fluxes differentiating the conservative vector. Earlier work by Cavalca et al. (2018) showed the robustness and accuracy of this implicit solver to predict inviscid flows over the airfoil and into the supersonic nozzle. Finally, the Gauss-Seidel (GS) iterative method was applied to solve the resultant sparse and large system of equations. These numerical schemes and methods were applied to solve the laminar flow over a flat plate. Afterwards, the numerical solution was validated and verified with the exact Blasius solution. From the results, the numerical simulations exhibited superior robustness of the implicit-defect correction scheme when compared with the explicit scheme for compressible flows. All numerical particularities and their implementations are detailed in this paper.

The Midpoint Upwind Scheme on the Bakhvalov-Shishkin Mesh for Parabolic Singularly Perturbed Problems

In this paper, the parabolic singularly perturbed convection-diffusion problem is discretized by using the backward Euler method in time and the midpoint upwind scheme on the Bakhvalov-Shishkin mesh in space. This method is shown to be first-order convergent in time and space. Finally numerical experiments confirm the theoretical results.

Morley FEM for the fourth-order nonlinear reaction-diffusion problems

Nonconforming Morley finite element method is applied to a fourth order nonlinear reaction-diffusion problems. After deriving some regularity results to be used subsequently in our error analysis, Morley FEM is employed to discretize in the spatial direction to obtain a semidiscrete problem. A priori bounds for the discrete solution are derived and with the help of an auxiliary problem, optimal error estimates are proved for the semidiscrete scheme. Based on backward Euler method, a completely discrete scheme is analysed and wellposedness of the discrete problem is discussed. A priori error bounds are derived. It is observed that constants do not depend exponentially on the inverse of a small parameter appeared as the coefficient in the fourth order term and all the results are derived when the initial data are in H02. Finally, some computational experiments are conducted to confirm our theoretical findings.

An energy-stable parametric finite element method for anisotropic surface diffusion

We propose an energy-stable parametric finite element method (ES-PFEM) to discretize the motion of a closed curve under surface diffusion with an anisotropic surface energy γ(θ) – anisotropic surface diffusion – in two dimensions, while θ is the angle between the outward unit normal vector and the vertical axis. By introducing a positive definite surface energy (density) matrix G(θ), we present a new and simple variational formulation for the anisotropic surface diffusion and prove that it satisfies area/mass conservation and energy dissipation. The variational problem is discretized in space by the parametric finite element method and area/mass conservation and energy dissipation are established for the semi-discretization. Then the problem is further discretized in time by a (semi-implicit) backward Euler method so that only a linear system is to be solved at each time step for the full-discretization and thus it is efficient. We establish well-posedness of the full-discretization and identify some simple conditions on γ(θ) such that the full-discretization keeps energy dissipation and thus it is unconditionally energy-stable. Finally the ES-PFEM is applied to simulate solid-state dewetting of thin films with anisotropic surface energies, i.e. the motion of an open curve under anisotropic surface diffusion with proper boundary conditions at the two triple points moving along the horizontal substrate. Numerical results are reported to demonstrate the efficiency and accuracy as well as energy dissipation of the proposed ES-PFEM.