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On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.

In this study, we consider the coupled nonlinear Schrödinger equation under the influence of the multiplicative time noise. The coupled nonlinear Schrödinger equation, which shows the complex envelope amplitudes of the two modulated weakly resonant waves in two polarisations and is used to describe the pulse propagation in high birefringence fibre, has several uses in optical fibres.query:Journal instruction requires a city for affiliations; however, these are missing in affiliation [6]. Please verify if the provided city are correct and amend if necessary. The underlying model is analyzed numerically and analytically as well. For the computational results, the proposed stochastic backward Euler scheme is developed and its consistency is derived in the mean square sense. For the linear stability analysis, Von-Neumann criteria is used, given proposed stochastic scheme is unconditionally stable. The exact optical soliton solutions are constructed with the help of the -model expansion technique, which provided us with the Jacobi elliptic function solutions that will explore optical solitons and solitary waves as well. The initial and boundary conditions are constructed for the numerical result by some optical soliton solutions. The 3D, 2D and corresponding contour plot are drawn for the different values of noise. Mainly, the comparison of results is shown graphically in 3D and line plots for some newly constructed solutions by selecting suitable parameters value.

Strong approximation errors of both finite element semi-discretization and spatio-temporal full discretization are analyzed for the stochastic Allen–Cahn equation driven by additive noise in space dimension d ≤ 3 . The full discretization is realized by combining the standard finite element method with the backward Euler time-stepping scheme. Distinct from the globally Lipschitz setting, the error analysis becomes rather challenging and demanding, due to the presence of the cubic nonlinearity in the underlying model. By introducing two auxiliary approximation processes, we propose an appropriate decomposition of the considered error terms and introduce a novel approach of error analysis, to successfully recover the convergence rates of the numerical schemes. The approach is original and does not rely on high-order spatial regularity properties of the approximation processes. It is shown that the fully discrete scheme possesses convergence rates of order O ( h γ ) in space and order O ( τ γ 2 ) in time, subject to the spatial correlation of the noise process, characterized by ‖ A γ - 1 2 Q 1 2 ‖ L 2 < ∞ , γ ∈ [ d 3 , 2 ] , d ∈ { 1 , 2 , 3 } . In particular, a classical convergence rate of order O ( h 2 + τ ) is reachable, even in multiple space dimensions, when the aforementioned condition is fulfilled with γ = 2 . Numerical examples confirm the previous findings.

This paper presents an abstract framework for the analysis of a numerical scheme designed for degenerate parabolic equations with time-dependent operators. We derive quasi-optimal error estimates for the approximation and establish sufficient conditions that guarantee the existence of a unique numerical solution. In our analysis, we use a finite element method for the spatial discretization and an implicit Euler scheme for the temporal discretization. Furthermore, applications with different operator choices are considered to illustrate the abstract theory. Finally, numerical experiments are carried out in order to validate the performance of the method and to corroborate our theoretical results.

Inthis paper, we discuss two first-order completely discrete schemes based on Backward Euler and its linearized variant methods for the 2D Sobolev equations with Burgers’ type nonlinearity. First, we derive some a priori estimates for the semi-discrete scheme, then a priori bounds for the fully discrete scheme are obtained for the backward Euler approximation. Use of discrete Gronwall’s Lemma and Stolz-Cesaro’s classical result for sequences show that these estimates for the fully discrete scheme are valid uniformly in time. Moreover, an existence of a global attractor of a discrete dynamical system is derived. Further, optimal a priori error bounds are established, which may depend exponentially on time. It is shown that these error estimates are uniform in time under a uniqueness condition. Moreover, as the coefficient of dispersion μ in − μ Δ u t tends to zero, both the semi-discrete and completely discrete Sobolev equations converge to the corresponding Burgers’ equation linearly with respect to μ . Finally, some numerical examples are established in support of our theoretical analysis.

The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BDF, and Crank–Nicolson schemes) are analyzed. In fact, error bounds are obtained that do not depend on the inverse of the viscosity in the case where the solution is sufficiently smooth. The bounds for the divergence of the velocity as well as for the pressure are optimal. The analysis is based on the use of a specific Stokes projection. Numerical studies support the analytical results.

This paper presents a dynamic homotopy technique that can be used to calculate a preliminary result for a power flow problem (PFP). This result can then be used as an initial estimate to efficiently solve the PFP using either the classical Newton-Raphson (NR) method or its fast decoupled version (FDXB) while still maintaining high accuracy. The preliminary stage for the dynamic homotopy problem is formulated and solved by employing integration techniques, where implicit and explicit schemes are studied. The dynamic problem assumes an initial condition that coincides with the initial estimate for a traditional iterative method such as NR. In this sense, the initial guess for the FPF is adequately set as a flat start, which is a starting for the case when this initialization is of difficult assignment for convergence. The static homotopy method requires a complete solution of a PFP per homotopy pathway point, while the dynamic homotopy is based on numerical integration methods. This approach can require only one LU factorization at each point of the pathway. Allocating these points properly helps avoid several PFP resolutions to build the pathway. The hybrid technique was evaluated for large-scale systems with poor conditioning, such as a 109,272-bus model and other test systems under stressed conditions. A scheme based on the implicit backward Euler scheme demonstrated the best performance among other numerical solvers studied. It provided reliable partial results for the dynamic homotopy problem, which proved to be suitable for achieving fast and highly accurate solutions using both the NR and FDXB solvers.

In this article, we mainly consider a first-order decoupling penalty method for the 2D/3D time-dependent incompressible magnetohydrodynamic (MHD) equations in a convex domain. This method applies a penalty term to the constraint “div u = 0,” which allows us to transform the saddle point problem into two small problems to solve. The time discretization is based on the backward Euler scheme. Moreover, we derive the optimal error estimate for the penalty method under semi-discretization with the relationship 𝜖 = O (Δ t ). Finally, we give abundant of numerical tests to verify the theoretical result and the spatial discretization is based on Lagrange finite element.

In this paper, some local and parallel finite element methods based on two-grid discretizations are proposed and investigated for the unsteady Navier-Stokes equations. The backward Euler scheme is considered for the temporal discretization, and two-grid method is used for the space discretization. The key idea is that for a solution to the unsteady Navier-Stokes problem, we could use a relatively coarse mesh to approximate low-frequency components and use some local fine mesh to compute high-frequency components. Some local a priori estimate is obtained. With that, theoretical results are derived. Finally, some numerical results are reported to support the theoretical findings.

This paper investigates the residual type a posteriori error estimators for a fully discrete approximation to the solution of the time-dependent Poisson–Nernst–Planck equations, which are widely used to describe the electrodiffusion of ions in biomolecular solutions. The backward Euler scheme is used for the discretization in time and the continuous, piecewise linear triangular finite elements are applied to the space discretization. The main results consist in building error estimators and deriving computable upper and lower bounds on the error estimators. Some numerical experiments confirm the theoretical predictions and show the reliability and efficiency of the error estimators.