Unconditional Stable Scheme Research Articles

An unconditional stable modified finite element methods for Maxwell’s equation in Kerr-type nonlinear media

The purpose of this paper is to construct a newly efficient finite element method which named as modified finite element method, for Maxwell’s equation in Kerr-type nonlinear media. Because the efficient unconditional stable alternative direction implicit method can’t extend to the Maxwell’s equations in Kerr-type nonlinear media (the difficulty will be discussed in section 3.1), we design the modified method to construct an efficient unconditional stable scheme for Maxwell’s equations in Kerr-type nonlinear media. Furthermore, combined with the direct method, we construct a more efficient direct modified scheme. Comparisons of the efficiency, stability, convergence rate for our modified method and classical explicit leapfrog method were done by theoretical analysis and numerical experiments. The modified method is unconditional stable and its efficiency is not less than leapfrog method.

Unconditional H2$$ {H}^2 $$‐stability of the Euler implicit/explicit SAV‐based scheme for the 2D Navier–Stokes equations with smooth or nonsmooth initial data

AbstractIn this article, we propose ‐unconditional stable schemes for solving time‐dependent incompressible Navier–Stokes equations with smooth or nonsmooth initial data, , . The ‐stability analysis is established by leveraging the scalar auxiliary variable (SAV) approach. When dealing with nonsmooth initial data, we utilize a limited number of iteration of the semi‐implicit scheme followed by the SAV scheme. The overall efficiency is greatly enhanced due to the minimal computational cost of the semi‐implicit scheme and the explicit treatment of the nonlinear term within the SAV approach. The proposed schemes investigate two types of scalar auxiliary variables: the energy‐based variable and the exponential‐based variable. Rigorous proofs of the ‐unconditional stability of both schemes have been provided. Notice that both proposed numerical schemes enjoy unconditional long time stability for smooth and nonsmooth initial data when . Numerical experiments have been conducted to demonstrate the theoretical results.

Robust and stable schemes for time fractional molecular beam epitaxial growth model using SAV approach

In this paper, we are interested in the time fractional molecular beam epitaxial growth models and its numerical solutions. Basically, two types of numerical schemes are proposed and analyzed: one is based on the Crank–Nicolson formula while the other is a kind of backward differentiation formula. The former is proved to be unconditional energy stable with a modified energy on the general temporal mesh, including the graded mesh and the uniform mesh. However, A drawback is its sensitivity to the coefficient and the time step size. The other, i.e., the backward differentiation based scheme, is more robust in term of this sensitivity. Its unconditional stability with respect to the modified energy is proved only for the uniform mesh. The key of these results is the use of a new approach to the time fractional derivative. The idea is to first split the time fractional derivative into the local part and the history part, then treat the two parts in different ways. This splitting allows to derive a nonlocal free energy dispassion for the continuous solution, and makes the construction of stable numerical schemes much easier. It is found that the well known L1, L1–CN, L1+–CN and L2 formula with the sum-of-exponentials approach for the local derivative term, combined with the extended auxiliary approach for the nonlinear and history terms result in unconditional stable schemes, which satisfy the modified energy dispassion at the discrete level. Finally, a series of numerical examples are provided to verify the efficiency of the proposed schemes.

An a priori error analysis for a projection based variational multiscale finite element method for Oseen problems in a time-dependent domain

Stability and error estimates for a projection based variational multiscale finite element scheme for Oseen problem in a time-dependent domain are derived in this paper. The use of Geometric Conservation Law (GCL) provides an unconditional stable scheme, whereas a restriction on the time-step needs to be imposed to obtain stability estimates independent of the mesh velocity when GCL is violated. Further, a priori error estimate is derived for the semi-discrete problem obtained with the backward Euler time discretization.

Thermal-WLP: A Transient Thermal Simulation Method Based on Weighted Laguerre Polynomials for 3-D ICs

Thermal issue becomes a key challenge in 3-D integrated circuits (3-D ICs) due to their highly integrated and compact design. To give a fast and accurate prediction of the transient temperature field, a new unconditional stable scheme for heat conduction equation is proposed in this paper. This method, thermal-weighted Laguerre polynomials (WLP), uses WLPs as basis functions and Galerkin's method as a testing procedure, to eliminate the time variables of the temperature field. Thus, a marching-on-in-order scheme can be obtained, which is more efficient on computation than those marching-on-in-time schemes of conventional numerical methods, especially for 3-D ICs with complex structures. Two examples of the transient thermal problems in 3-D ICs are presented, and the numerical results show the accuracy and efficiency of the proposed method.

A new family of fourth-order locally one-dimensional schemes for the three-dimensional wave equation

In this paper, we present a new family of locally one-dimensional (LOD) schemes with fourth-order accuracy in both space and time for the three-dimensional (3D) acoustic wave equation. It is well-known that high order explicit schemes offer efficient time steppings but with restrictive CFL conditions; implicit discretization gives unconditional stable schemes but with inefficient time steppings. Our LOD schemes can be seen as a compromise between explicit and implicit schemes, in the way that our schemes have more relax CFL conditions compared with traditional explicit schemes and only require solutions of tridiagonal systems of linear algebraic equations, which is more efficient than implicit schemes. Moreover, the new scheme is four-layer in time and three-layer in space, so that boundary conditions can be imposed in the classical way. The computations of the initial conditions for the three intermediate time layers are explicitly constructed. Furthermore, the stability condition is derived and the restriction for the time step is given explicitly. Finally, numerical examples are completed to show the performance of our new method.

Unconditionally Explicit Stable Difference Schemes for Solving Some Linear and Non-Linear Parabolic Differential Equation

We present the numerical method for solution of some linear and non-linear parabolic equation. Using idea [1], we will present the explicit unconditional stable scheme which has no restriction on the step size ratio k/h2 where k and h are step sizes for space and time respectively. We will also present numerical results to justify the present scheme.

A STUDY ON UNCONDITIONALLY STABLE EXPLICIT DIFFERENCE SCHEMES FOR THE VARIABLE COEFFICIENTS PARABOLIC DIFFERENTIAL EQUATION

In (2), we have presented some new algorithms for solving the linear variable coefficients parabolic differential equations. The proposed scheme is stable without any restriction to step-sizes of space and time. The scheme is required the condition of step size ratio k h2,→ 0 as k,h → 0 in the convergence, where k and h are step sizes for space and time respectively. In this paper, we will present the explicit unconditional stable scheme which has no restriction on the step size ratio k h2 in the convergence. We will also present analysis for the present scheme.

Wave equation simulation on manifold by unconditional stable schemes

To predict the wave propagation in a given region over time, it is often necessary to find the numerical solution for wave equation. With the techniques of discrete differential calculus, we propose two unconditional stable numerical schemes for simulation wave equation on the space manifold and the time. The analysis of their stability and error is also accomplished.

Two Unconditional Stable Schemes for Simulation of Heat Equation on Manifold Using DEC

To predict the heat diffusion in a given region over time, it is often necessary to find the numerical solution for heat equation. However, the computational domain of classical numerical methods are limited to flat spacetime. With the techniques of discrete differential calculus, we propose two unconditional stable numerical schemes for simulation heat equation on space manifold and time. The analysis of their stability and error is accomplished by the use of maximum principle.

Nonstandard numerical methods for a mathematical model for influenza disease

In this paper we construct and develop two competitive implicit finite difference scheme for a deterministic mathematical model associated with the evolution of influenza A disease in human population. Qualitative dynamics of the model is determined by the basic reproduction number, R 0 . Numerical schemes developed here are based on nonstandard finite difference methods. Our aim is to transfer essential properties of the continuous model to the discrete schemes and to obtain unconditional stable schemes. The proposed numerical schemes have two fixed points which are identical to the critical points of the continuous model and it is shown that they have the same stability properties. Numerical simulations with different initial conditions, parameters values and time step sizes are developed for different values of parameter R 0 , convergence to the disease free equilibrium point when R 0 < 1 and to the endemic equilibrium point when R 0 > 1 are obtained independent of the time step size. These numerical integration schemes are useful since can reproduce the dynamics of original differential equations.