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The multi-harmonic balance method is widely applied to obtain the forced responses of nonlinear systems undergoing rubbing problems. Despite large-scale time savings compared with the time marching method, it suffers from the complicated derivations of the Jacobian matrix. To solve this problem, this paper focuses on applying the automatic differentiation frame to the multi-harmonic balance method to implement the nonlinear vibration analysis of systems subjected to the rub phenomena. By establishing computational graph and utilizing the automatic differentiation process, tedious works such as the derivations of the complicated analytical expressions of the Jacobian matrix are avoided, which guarantees the efficiency and applicability of the presented method. A single-degree-of-freedom system with nonlinear force in the form of cubic is used to verify the accuracy of the method, and numerical analysis results reveal that the method is accurate enough compared with the time marching method. Furthermore, for the purpose of application, the responses of two common friction models, which are of great concern in some practical engineering fields, including a two-degree-of-freedom system containing a friction damper and a rotor disk system with circumferential rubbing, are obtained utilizing the presented approach. The influences of several model parameters on their responses are investigated as well. Numerical investigations demonstrate that the automatic differential solution framework developed in this research for solving nonlinear vibration equations has high accuracy and eliminates the need for a complicated partial derivative analytical formula derivation.

External corrosion is one of the leading causes of pipe failure in the oil and gas industry. In this study, a Bayesian belief network (BBN) model has been developed using corrosion rate (CR) data obtained from experimental test results and analytical burst failure models. The BBN model for CR was coupled with a time marching simulation to obtain corrosion defects and quantify failure pressure capacity. Finally, in a reliability framework, the failure pressure capacity was coupled with operating pressure to obtain the probability of failure. Furthermore, the developed BBN model was used to perform a parametric study to identify the critical parameters for the CR. The outcome of the study indicated that the proposed BBN model can be useful to integrate experimental and analytical models to derive reliability of a pipeline operating under various conditions.

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.

The time-space-domain finite-difference method (FDM) is widely used in forward modeling of wave equations. Conventional explicit FDMs (EFDMs) (with high order in space and the second order in time) would result in apparent temporal and spatial dispersion for high frequencies and large time steps. Moreover, the saturation effect of Taylor expansion seriously restricts the improvement of bandwidth coverage and efficiency of EFDMs. We have developed a variable-order optimization scheme of the implicit FDM (IFDM) to improve the computational efficiency and numerical accuracy of forward modeling. Then, we applied time-dispersion transforms (TDTs) to filter out the temporal dispersion generated by the second-order temporal approximations. Our method greatly alleviates the saturation effect of the high-order spatial finite-difference (FD) operators. Dispersion analysis indicates that the optimized coefficients of the IFDM based on the Remez algorithm can achieve the widest bandwidth (close to the Nyquist wavenumber), which corresponds to the shortest length of the spatial FD operator under a given error threshold. Our method has great potential to approach the highest spectral accuracy but with minimal increase in computational cost. Numerical experiments indicate that the combination of the variable-order optimization of IFDM and TDTs can significantly reduce numerical dispersion. Compared with traditional methods, our scheme is more advantageous for numerical simulation of large-scale geologic models because it has the least amount of calculation burden under the same accuracy requirements.

In this paper, we present a fully discrete analysis of an H(div)-conforming DG method with semi-implicit time-marching for the evolutionary incompressible Navier–Stokes equations. We use a semi-implicit time-discrete scheme in which the convection velocity is treated explicitly for the convection term. A stability analysis and a priori error estimates are given, in which the constants are independent of the negative powers of the viscosity. For inf-sup stable H(div)-conforming finite element pairs $BDM_k/P_{k-1}$ and $RT_k/P_k$, the rate of convergence $k+1/2$ is proved for the $L^2$ error of the velocity in the case of $\nu \leq C h$, where $k$ is the degree of the polynomials in the velocity approximation. In particular, for the inf-sup stable finite element pair $RT_k/P_k$, the convergence rate of the pressure is also $k+1/2$ when $\nu \leq C h$. The numerical experiments verify the analytical results.

A new blade force model is coupled to quasi-one dimensional Euler equations for a variable geometry flowpath. After analytical inclusion of the blade force, the flow equations take a strictly one-dimensional form with specific expressions of the convective flux and blade load source terms. Regardless of the flow turning, that is simply achieved by the load source term as an explicit function of the blade camber, the new form describes a perfect analogy between the average flow inside a blade passage and strictly one-dimensional flows, especially concerning wave propagation. This property allows capture of passage choking and shocks. Other types of shock more important for turbomachinery analysis, like leading edge strong shocks in compressors and trailing edge weak shocks in choked turbines, are modelled by properly matching the new set of equations inside blade regions with the standard quasi-one dimensional equations outside. Upon specification of viscous losses and subsonic deviations fitted from experimental results, the model predicts the choke mass flow of a transonic compressor stage (NASA stage 37) at a 0.1% to 0.4% accuracy both in the absence and in the presence of the leading edge shock. This result supports the effectiveness of the leading edge shock model. The accuracy on choke mass flow would decrease to around 1% if empirical input was specified from open-literature experimental correlations. The model captures the typical trend of exit angle with total pressure ratio for a choked turbine (NASA Lewis two-stage). This result involves satisfactory prediction of not only choke mass flow, but also trailing edge shock loss and supersonic deviation. The complete turbine operational map in terms of shaft torque and pressure ratio is also re-obtained with noticeable accuracy except in strong off-design conditions, where experimental correlations likely fail.

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.

An efficient method is proposed in this study for solving the semi-analytical solutions of periodic responses of nonlinear oscillators. The basic ideas come from the fact that any periodic response can be described by Fourier series. By transforming the Fourier series into a system of harmonic oscillators, we thus establish an efficient numerical scheme for tracking the periodic responses, as long as a synchronized motion can be achieved between the system of harmonic oscillators and the nonlinear oscillators considered. The presented method can be implemented by conducting time marching integration only, but it is capable of providing semi-analytical solutions straightforwardly. Different from some widely used methods such as harmonic balance method and its improved forms, this method can solve solutions involving high order harmonics without incorporating any tedious derivations as it is totally a numerical scheme. Several typical oscillators with smooth as well as nonsmooth nonlinearities are taken as numerical examples to test the validity and efficiency.

In this paper, two explicit time-marching techniques are discussed for the solution of hyperbolic models, which are based on adaptively computed parameters. In both these techniques, time integrators are locally and automatically evaluated, taking into account the properties of the spatially/temporally discretized model and the evolution of the computed responses. Thus, very versatile solution techniques are enabled, which allows computing highly accurate responses. Here, the so-called adaptive [Formula: see text] method is formulated based on the elements of the adopted spatial discretization (elemental formulation), whereas the so-called adaptive [Formula: see text] method is formulated based on the degrees of freedom of the discretized model (nodal formulation). In this context, each adaptive procedure can be better applied according to the specific features of the focused spatial discretization technique. At the end of the paper, numerical results are presented, illustrating the excellent performance of the discussed adaptive formulations.

A novel hybrid finite-element time-domain (FETD) algorithm is proposed to solve electromagnetic modeling with fine structures. This algorithm is based on the unconditionally stable FETD (US-FETD) and the traditional FETD methods. The US-FETD is used in the subdomain containing all fine grids and parts of coarse grids adjacent to the fine grids. The traditional FETD is applied in the other subdomain. A compensation and correction scheme is used to update the fields on the subdomain boundary. The hybrid FETD can make the explicit time marching unconditionally stable with a uniform time step determined by the coarse grids in the entire computational domain. In addition, the unstable modes of the entire system in the hybrid FETD can be effectively obtained by solving the partial solutions of the eigenvalue problem in the US-FETD subdomain. In comparison with the US-FETD in the entire computational domain, the eigenvalue analysis costs less memory and time. Numerical results certify the high efficiency of the hybrid FETD.

In this paper, we develop the fully discrete local discontinuous Galerkin (LDG) methods coupled with the implicit-explicit (IMEX) multistep time marching for solving the nonlinear Cahn-Hilliard equation. To be more specific, we rewrite the Cahn-Hilliard equation in a novel form by adding and subtracting a “linear” term. Then we discretize the spatial derivatives with the LDG methods and the temporal derivative with the IMEX multistep method. Finally, a series of numerical experiments are given to verify the accuracy, efficiency and validness of proposed method.

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.

espanolUtilizando el analisis de von Neumann como herramienta teorica,se desarrolla un analisis sobre las condiciones de estabilidad de algunosmetodos explicitos de avance en tiempo, en combinacion con la discretizacion espacial Local Discontinuous Galerkin (LDG) por sus siglas en ingles y aproximaciones de alto orden. La constante de estabilidad CFL (Courant-Friedrichs-Lewy) se estudia en funcion de los parametros del metodo LDG y el grado de aproximacion. Se realiza una serie de experimentos numericos para validar los resultados teoricos. EnglishUsing the von Neumann analysis as a theoretical tool, an analysisof the stability conditions of some explicit time marching schemes, in combinationwith the spatial discretization Local Discontinuous Galerkin (LDG)and high order approximations, is presented. The stability constant, CFL(Courant-Friedrichs-Lewy), is studied as a function of the LDG parametersand the approximation degree. A series of numerical experiments is carriedout to validate the theoretical results.

Integrated environmental modelling (IEM) couples environmental models, geoprocessing algorithms, and data together to solve complex environmental problems. There are two prominent modelling frameworks for IEM: component-based and service-oriented frameworks. The former one has been widely employed in local computer environments, and has its advantage in having concrete and community-wide tools. The latter one can take advantage of Web technologies for integrating distributed environmental models. However, the complex interaction between environmental models such as time marching models is still a challenge for integrating models on the Web. This paper suggests to take the best of both frameworks. Models are shared on the Web as WebSocket services. A hybrid approach to leverage OpenMI and WebSocket for coupling components and services is proposed to provide a flexible IEM environment. The result illustrates the applicability of the approach and demonstrates the advantage of WebSocket over the traditional HTTP approach in model execution.

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.

A time-marching throughflow method for the off-design performance analysis of axial compressors is described. The method is based on the Euler equations, and a new inviscid blade force model is proposed in order to achieve desired flow deflection. The flow discontinuity problems at the leading and trailing edges are tackled by automatic correction of blade mean surface using cubic spline interpolation. Empirical loss models have been integrated into the throughflow model in order to simulate the viscous force effects in the real three-dimensional flow. Two test cases have been presented to validate the throughflow model, including the transonic fan rotor – NASA Rotor 67 working at a near-peak-efficiency point and a 1.5-stage high-speed axial compressor with inlet guide vane operating at 68% nominal speed. Reasonable flow parameters distributions have been obtained in the Rotor 67 fan calculating results, and accurate overall performance characteristics have also been predicted at the strong off-design condition for the 1.5-stage axial compressor. The CPU time of both cases cost less than one minute at one operating point. The results indicate that the developed time-marching throughflow model is effective and efficient in the turbomachinery performance analysis.

The time step of the traditional mixed time-domain finite element method (TM-TDFEM) based on E-B co-simulation is small due to the fine features, which causes tremendous number of steps for time marching, increases rounding errors and raises computational costs. In this paper, a stable mixed TDFEM (SMTDFEM) is implemented, which can reach unconditional stability by introducing the filtering spatial unstable modes method. Numerical results illustrate the accuracy and efficiency of the proposed method.

In this work, we develop an efficient energy stable scheme for the hydrodynamics coupled phase-field surfactant model with variable densities. The thermodynamically consistent model consists of two Cahn–Hilliard–type equations and incompressible Navier–Stokes equation. We use two scalar auxiliary variables to transform nonlinear parts in the free energy functional into quadratic forms, and then they can be treated efficiently and semi-implicitly. A splitting method based on pressure stabilization is used to solve the Navier–Stokes equation. By some subtle explicit-implicit treatments to nonlinear convection and stress terms, we construct a first-order energy stable scheme for the two-phase system with soluble surfactants. The developed scheme is efficient and easy-to-implement. At each time step, computations of phase-field variables, the velocity and pressure are decoupled. We rigorously prove that the proposed scheme is unconditionally energy stable. Numerical results confirm that our scheme is accurate and energy stable.

Two new numerical methods based on Haar wavelet collocation and meshless collocation are proposed for solution of parabolic type of time marching interface models. This work is extension of the earlier work (Aziz et al., 2018) from one-dimensional elliptic type interface models to one-dimensional parabolic type interface models. Performance of both the methods is checked in terms of L∞ error norm. Comparison of the proposed methods with the existing methods validates superiority of the methods.

This study focuses on the development of time-marching procedures for efficient and accurate fluid-thermal-structural analysis with time-accurate computational fluid dynamics. The developed procedu...

In the community of computational fluid dynamics, pressure Poisson equation with Neumann boundary condition is usually encountered when solving the incompressible Navier–Stokes equations in a segre...

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.

This paper presents a novel sequential-explicit coupling technique to solve hydromechanical problems in poroelastic media. The proposed approach combines appropriate coupling terms defined in a rate form and an automatic time-stepping scheme that controls the temporal discretization error through a single parameter. Such technique has been implemented in a framework that manages sequential simulations, exchanges of information between models and adapts the time marching in a multirate approach. We present several numerical tests in which we evaluate the new scheme’s stability, accuracy, and performance through comparisons with the fully implicit solution. We observe that the proposed technique is stable, robust and accurate, and can be used in engineering applications for large scale and long time durations.

This paper addresses the temporal dynamics of the time marching when this approach is used for determining the steady solution of advection–diffusion problems. The time-step size, δt, is the main p...

In this paper, we present a stable method for simulating passive distributed networks connected to linear and nonlinear devices using the time marching (TM) method. The method is based on the use of the Z -transform for analyzing and obtaining the TM equations, and the application of cepstrum for preventing the apparition of unstable poles in the Z -domain counterparts of the TM equations. The efficiency of the proposed method is demonstrated by comparing its results with simulations obtained using the following: 1) a full-wave FDTD approach, and 2) CST Cable Studio.

Chapter 8 provides recent developments and mathematical treatments of the extended method of fundamental solutions for the general two-dimensional heat transfer problems. For the static case, the analog equation method is used to convert the original static governing equations into equivalent Poisson's equations, whose homogeneous and particular solutions can be respectively modeled by the standard method of fundamental solutions and the radial basis function interpolation. The strong-form satisfaction of the original static governing equations and the specific boundary conditions at interior and boundary collocations can be used to determine all unknowns in the solving system. Such a unified solving framework offers the readers a potential to extend the present meshless strategy further to transient heat transfer in heterogeneous material by the help of time marching treatment of the transient response of temperature.

We study the numerical anisotropy existent in compact difference schemes as applied to hyperbolic partial differential equations, and propose an approach to reduce this error and to improve the stability restrictions based on a previous analysis applied to explicit schemes. A prefactorization of compact schemes is applied to avoid the inversion of a large matrix when calculating the derivatives at the next time level, and a predictor-corrector time marching scheme is used to update the solution in time. A reduction of the isotropy error is attained for large wave numbers and, most notably, the stability restrictions associated with MacCormack time marching schemes are considerably improved. Compared to conventional compact schemes of similar order of accuracy, the multidimensional schemes employ larger stencils which would presumably demand more processing time, but we show that the new stability restrictions render the multidimensional schemes to be in fact more efficient, while maintaining the same dispersion and dissipation characteristics of the one dimensional schemes.

The numerical approximations for the coupled Cahn–Hilliard system describing the phase separation of the copolymer and homopolymer mixtures are considered in this paper. To develop easy to implement time marching schemes with unconditional energy stabilities, we use the Scalar Auxiliary Variable (SAV) approach for achieving two efficient, decoupled, and linear numerical schemes, where a new scalar auxiliary variable is introduced to reformulate the model. The schemes lead to decoupled linear equations with constant coefficients at each time step, and their unique solvability and unconditional energy stabilities are proved rigorously. Numerical examples are performed to demonstrate the accuracy and energy stability of the proposed schemes, and numerous benchmark simulations are also presented to show a variety of morphologies of pattern formations of the copolymer and homopolymer mixtures.

In this paper, a truly explicit and truly self-starting time-marching procedure is proposed for structural dynamics and wave propagation analyses. The novel technique can be subdivided into three different levels, providing distinct time-marching formulations that can be applied separately or considering a mixed adaptive approach. In this context, at levels 1 or 2, the new technique provides simpler and more accurate/stable formulations than their equivalent counterparts in standard time-marching techniques. At level 3, the novel approach enables a mixed self-adjustable space/time co-joint solution procedure, providing a highly effective time integration technique for wave propagation analyses. Numerical results are presented along the manuscript, illustrating the excellent performance of the proposed multi-level time-marching technique.

In this paper, we develop a series of efficient numerical schemes to solve the phase field model for homopolymer blends. The governing system is derived from the energetic variational approach of a total free energy, that consists of a nonlinear logarithmic Flory–Huggins potential, and a gradient entropy with a concentration-dependent de-Gennes type coefficient. The main challenging issue to solve this kind of models numerically is about the time marching problem, i.e., how to develop suitable temporal discretizations for the nonlinear terms in order to preserve the energy stability at the discrete level. We solve this issue in this paper, by developing the first and second order temporal approximation schemes based on the “Invariant Energy Quadratization” method, where all nonlinear terms are treated semi-explicitly. Consequently, the resulting numerical schemes lead to a symmetric positive definite linear system to be solved at each time step. The unconditional energy stabilities are further proved. Various numerical simulations of 2D and 3D are presented to demonstrate the stability and the accuracy of the proposed schemes.

In this study, a numerical model based on shallow water equations (SWE) is developed. An explicit predictor–corrector approach is adopted for the time marching process. Using the leap-frog formulae, the three unknowns in SWE, which are the water depth h , and the water fluxes uh , vh , are firstly estimated directly by their values and their spatial derivatives in the previous time step. Then they are corrected by the Crank-Nicolson formulation. The spatial derivatives of h , uh and vh for the further time marching processes are calculated by using the Weighted Least Square (WLS) local polynomial approximation, which is a kind of meshless method. This model is applied to the simulations of dam break flows and tidal currents.

In this work, an unconditionally stable family of time marching procedures is proposed, in which the time integration parameters of the method are locally computed, considering the properties of the model. Thus, the technique adapts to the characteristics of the problem, enabling enhanced analyses. Here, adaptive dissipation control is explored, allowing spurious modes to be more effectively dissipated. The proposed technique is second-order accurate, L -stable, truly self-starting and very easy to implement. Numerical results are presented along the paper, illustrating the good performance of the novel single-step technique.

In this paper, an optimized approach is proposed to enhance the performance of combined explicit–implicit time-domain analyses. In this context, an entirely automated explicit-implicit adaptive time-marching procedure is discussed as well as an optimization algorithm is introduced to compute the adopted time-step value of the analysis, so that the amount of explicit and implicit elements occurring along the model may be optimally provided, in terms of computational efficiency. The proposed formulation is very effective, allowing evaluating highly accurate responses considering much reduced computational efforts. At the end of the manuscript, numerical applications are presented, illustrating the excellent performance of the proposed formulation.

In this paper, we develop an unconditionally energy stable, second-order accurate time marching scheme for solving a penalized Allen–Cahn type Ohta–Kawasaki phase-field model for diblock copolymers, where the total free energy of the system consists of the double-well potential, the nonlocal Ohta–Kawasaki free energy functional, and a penalization potential to enforce the conservation of the modified volume approximately. The developed scheme combines the SAV (scalar auxiliary variable) approach with the stabilization technique, where a crucial linear stabilization term is added to enhance the stability while using the large time steps. The scheme is very easy-to-implement and one only needs to solve two decoupled elliptic equations with constant coefficients at each time step. We further prove the unconditional energy stability of the scheme rigorously and demonstrate the stability and the accuracy of the developed scheme numerically through simulating numerous numerical examples in 2D and 3D.

In this paper, we expand the classical phase-field model for diblock copolymers (BCPs) to the magnetic-coupled case, where the free energy of the system is modeled as the combination of the nonlocal Ohta-Kawasaki free energy and the classical Ginzburg-Landau formalism for ferromagnetic ordering. To solve the model, we develop an efficient, second-order time marching scheme by adopting the recently developed stabilized-SAV (Scalar Auxiliary Variable) approach. The energy stability of the scheme is proved and several numerical examples are presented to show that the morphology patterns are strongly influenced by the magnetic field.

The purpose of this chapter is to present some properties inherent in the governing equations listed in Sect. 1.1. Most problems in the geophysical sciences, including atmospheres, oceans, seas, and lakes, involve solving so-called time marching problems. Typically, the state of the fluid in question is known at one specific time. As postulated by Bjerknes (Meteor Z 21:1–7, 1904) in his comments on weather forecasting (see quote in the preface), the aim is then to predict the state of the fluid at a later time. This is done by solving the governing equations listed in Sect. 1.1. Such problems are known in mathematics as initial value problems.