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In this paper, three novel truly-explicit time-marching techniques are proposed for the solution of hyperbolic models. In these techniques, locally computed time-integration parameters are applied, which are defined taking into account the properties of the spatially/temporally discretised model and the evolution of the computed responses. These adaptive time integrators allow introducing enhanced numerical damping into the analysis, reducing spurious non-physical oscillations that occur due to the excitation of spatially unresolved modes. The proposed adaptive techniques are formulated based on second-, third-, and fourth-order accurate time-marching approaches, providing solution algorithms with different complexities. At the end of the paper, numerical results are presented, illustrating the good performance of the proposed procedures.

In this paper, η-order accurate truly-explicit time-marching procedures are proposed for dynamic analyses. In these new techniques, just η–1 stiffness matrix–vector operations are required for solution, per time-step, providing more efficient approaches than standard high-order time-marching procedures, which demand η matrix–vector operations per time-step. In addition, as illustrated along this work, the novel formulations are also more accurate than equivalent standard techniques, providing very effective time-domain solution procedures. Three truly-explicit time-marching procedures are discussed here, describing second-, third- and fourth-order accurate methodologies.

In this paper, a novel time-marching procedure is proposed for dynamic analysis. The proposed new technique is fourth-order accurate for undamped models; third-order accurate for numerically damped models; and second-order accurate for physically damped models. The method is very straightforward to implement and to apply, allowing providing higher-order accurate time-domain solutions taking into account basically the same computational effort of standard single-step first- or second-order accurate time-marching procedures. Numerical examples are presented at the end of the paper, illustrating the excellent accuracy of the novel approach.

We present an adaptive time stepping technique, based on a dual-time stepping scheme together with a RK/implicit smoother for physical time marching. This method is found to be very efficient for transient problem characterized by a large variations of time scales during the physical time and hence a wide range of required physical time steps. We discuss the method and present several examples of compressible flow problems.

Generally, solving linear systems from finite difference alternating direction implicit scheme of two-dimensional time-space fractional differential equations with Gaussian elimination requires $\mathcal {O}\left ({NM}_{1}M_{2}\left ({M_{1}^{2}}+{M_{2}^{2}}+NM_{1}M_{2}\right )\right )$ complexity and $\mathcal {O}\left ({N{M_{1}^{2}}{M_{2}^{2}}}\right )$ storage, where N is the number of temporal unknown and M1, M2 are the numbers of spatial unknown in x, y directions respectively. By exploring the structure of the coefficient matrix in fully coupled form, it possesses block lower-triangular Toeplitz structure and its blocks are block-dense Toeplitz matrices with dense-Toeplitz blocks. Based on this special structure and cooperating with time-marching or divide-and-conquer technique, two fast solvers with storage $\mathcal {O}\left ({NM}_{1}M_{2}\right )$ are developed. The complexity for the fast solver via time-marching is $\mathcal {O}\left ({NM}_{1}M_{2}\left (N+\log \left (M_{1}M_{2}\right )\right )\right )$ and the one via divide-and-conquer technique is $\mathcal {O}\left ({NM}_{1}M_{2}\left (\log ^{2} N+ \log \left (M_{1}M_{2}\right )\right )\right )$. It is worth to remark that the proposed solvers are not lossy. Some discussions on achieving convergence rate for smooth and non-smooth solutions are given. Numerical results show the high efficiency of the proposed fast solvers.

In this article, a meshless spectral radial point interpolation method is proposed for the numerical solutions of a class of time-fractional advection–diffusion–reaction equations. The current approach utilizes meshless shape functions, having Kronecker delta function property, for approximation of spatial operators. Forward difference along with quadrature formula is used for tempered fractional derivative approximation in the framework of implicit time marching scheme. Assessment of the proposed method is made by applying to various concrete test problems having constant and variable coefficients. Approximation and function reproduction quality are measured via $${E}_{\infty }$$ , $${E}_{2}$$ and $${E}_{\text {rms}}$$ error norms. Comparison of simulated results is also made with available exact solutions as well as earlier reported works. Stability analysis of the proposed method is thoroughly discussed and computationally affirmed.

In this paper, we shall present two fully-discrete local discontinuous Galerkin methods, coupled with multi-step implicit–explicit time discretization up to second order, for solving the two-dimensional incompressible miscible displacement problem. To avoid the solving of nonlinear algebraic systems, the extrapolation linearization is adopted to diffusion–dispersion tensor. Under weak temporal-spatial conditions, the optimal error estimates in $$L^{\infty }(L^{2})$$ norm for both concentration and velocity are derived. Numerical experiments are also given to demonstrate the theoretical results.

In this work, adaptive explicit and implicit time marching techniques are proposed to analyze nonlinear dynamic models. In these time marching adaptive procedures, the adopted time integration parameters modify themselves along the solution process, in accordance with the varying properties and results of the model. Thus, the time integrators are locally evaluated, assuming different values along the spatial and temporal discretizations, enabling more accurate and efficient solution techniques. Numerical results are presented along the manuscript, illustrating the good performance of the proposed space/time adaptive explicit and implicit time marching strategies for nonlinear analyses.

In this paper, two new analytical techniques named max–min approach (MMA) and iteration perturbation method (IPM) are applied for solving nonlinear equations of two oscillatory systems. One case consists of a mass grounded by linear and nonlinear springs in series and the other case is the rigid rod rocks on a circular surface. The main aim of the work is obtaining highly accurate analytical solutions for nonlinear free vibration of conservative oscillations and investigates the dynamic behavior of the systems. Results reveal that the nonlinear frequency of oscillatory systems remarkably affected with the initial conditions. In contrast to the time marching solution results, the present analytical methods are effective and convenient. It is predictable that the MMA and the IPM can apply for various problems in engineering specially vibration equations.

It is shown that the isotropic wave-like multidimensional spatial stencils combined with linear multistep and Runge-Kutta time marching schemes provide more favorable stability restrictions for advective initial-value problems. Under certain conditions the maximum allowable time step can be doubled compared to using conventional spatial stencils. Consequently, this paper shows that the multidimensional optimizations of spatial schemes, involving more grid points, are not inherently less efficient in terms of the processing time. Three numerical tests solving the two and three dimensional advection equations are carried out to experiment the stability restrictions found in the previous sections.

New time marching algorithms for numerical solution of semi-linear parabolic equations are described. They are based on the approximation method proposed by the first author. An important feature of the algorithms is that they are both explicit and stable under mild restrictions to the time step, which come from the non-linear part of the equation.

In this paper, we have proposed a time marching intregral equation method which does not have the limitation of the time linearized integral equation method in that the latter method can not satisfactorily simulate the shock-wave motions. Firstly, a model problem—one dimensional initial and boundary value wave problem is treated to clarify the basic idea of the new method. Then the method is implemented for 2-D and 3-D unsteady transonic flow problems. The introduction of the concept of a quasi-velocity-potential simplifies the time marching integral equations and the treatment of trailing vortex sheet condition. The numerical calculations show that the method is reasonable and reliable.

Numerical methods for time stepping the Cahn-Hilliard equation are given and discussed. The methods are unconditionally gradient stable, and are uniquely solvable for all time steps. The schemes require the solution of ill-conditioned linear equations, and numerical methods to accurately solve these equations are also discussed.