Fourth-order Time Integration Research Articles

On the effect of temporal error in high-order simulations of unsteady flows

The present paper focuses on the effect that temporal error has on unsteady solutions modeled with a high-spatial-order, finite-element, streamline upwind/Petrov-Galerkin solver. Six L-stable, first- through fourth-order time-integration methods are discussed and exercised on two canonical test cases: the convecting isentropic vortex and the two-dimensional circular cylinder in crossflow. All results are shown on grids composed of P2 (third-order) elements. The isentropic vortex case is used to verify temporal order of accuracy, assess the relative split between dissipative and dispersive errors for each method with various time steps, and quantify the relative computational cost of each method. The results from the two-dimensional circular cylinder in crossflow simulations at various Reynolds numbers are compared to experimental results and accepted numerical results for vortex shedding frequency (Strouhal number), mean drag coefficient, and RMS lift and drag coefficient. The convergence of those parameters with respect to time step size is shown. These results highlight the benefits of high-order (greater than second order) time-integration methods and the need for time-step convergence studies on unsteady simulations.

Efficient high-order compact exponential time differencing method for space-fractional reaction-diffusion systems with nonhomogeneous boundary conditions

This paper introduces an efficient unconditionally stable fourth-order method for solving nonlinear space-fractional reaction-diffusion systems with nonhomogeneous Dirichlet boundary conditions on bounded domains. The proposed method is based on a compact improved matrix transformed technique for fourth-order spatial approximation and exponential time differencing approximation for fourth-order time integration. The main advantage of the improved matrix transfer technique is that it leads to a system of ordinary differential equations with spatial discretization matrix raised to the desired fractional order. The key benefit of the fourth-order exponential integrator is that it can be implemented with essentially the same computational complexity as the backward Euler method by utilizing a partial fraction splitting technique in which it is just required to solve two backward Euler-type well-conditioned linear systems at each time step by computing LU decomposition of spatial discretization matrix once outside the time loop. Linear stability analysis and various numerical experiments are also performed to demonstrate stability and accuracy of the proposed method. Moreover, calculation of local truncation error and an empirical convergence analysis show the fourth-order accuracy of the proposed method.

Fourth-order two-stage explicit exponential integrators for time-dependent PDEs

Among the family of fourth-order time integration schemes, the two-stage Gauss–Legendre method, which is an implicit Runge–Kutta method based on collocation, is the only superconvergent. The computational cost of this implicit scheme for large systems, however, is very high since it requires solving a nonlinear system at every step. Surprisingly, in this work we show that one can construct and prove convergence results for exponential methods of order four which use two stages only. Specifically, we derive two new fourth-order two-stage exponential Rosenbrock schemes for solving large systems of differential equations. Moreover, since the newly schemes are not only superconvergent but also fully explicit, they turn out to be very competitive compared to the two-stage Gauss–Legendre method as well as other fourth-order time integration schemes. Numerical experiments are given to demonstrate the efficiency of the new integrators.

Conservative fourth-order time integration of non-linear dynamic systems

Abstract An energy conserving time integration algorithm with fourth-order accuracy is developed for dynamic systems with nonlinear stiffness. The discrete formulation is derived by integrating the differential state-space equations of motion over the integration time increment, and then evaluating the resulting time integrals of the inertia and stiffness terms via integration by parts. This process introduces the time derivatives of the state space variables, and these are then substituted from the original state-space differential equations. The resulting discrete form of the state-space equations is a direct fourth-order accurate representation of the original differential equations. This fourth-order form is energy conserving for systems with force potential in the form of a quartic polynomial in the displacement components. Energy conservation for a force potential of general form is obtained by addition of a higher order secant-type correction term. The formulation leads to a consistent representation of the motion within a time increment corresponding to cubic Hermite interpolation in time. This in turn leads to excellent phase representation with only a small fourth-order error, permitting integration of oscillatory systems with only a few integration points per period. Three numerical examples demonstrate the high accuracy of the algorithm.

Closure to “Evaluation of Explicit Numerical Solution Methods of the Muskingum Model” by Ali R. Vatankhah

Closure to “Evaluation of Explicit Numerical Solution Methods of the Muskingum Model” by Ali R. Vatankhah

Numerical analysis of various artificial damping schemes in a three-dimensional numerical wave tank

Various artificial damping schemes for progressive water waves were evaluated in a three-dimensional numerical wave tank (3D NWT). Five types of damping schemes were applied to the end of a computational domain and their damping capabilities were compared to find an optimum artificial damping scheme. The 3D NWT was based on a nine-node, bi-quadratic, higher-order boundary element method (HOBEM). A semi-Lagrangian approach known as the Mixed-Eulerian–Lagrangian (MEL) method was used to capture the nonlinear free surface fluctuation. A Runge–Kutta fourth-order time integration scheme was used for time marching to produce a time history of water surface elevations. For efficient damping performance, four different ramping functions were introduced and applied to the damping coefficients, and their influences were evaluated in order to select an optimal ramping shape along the artificial damping zone. The conservation of total energy in the computational domain was ensured to verify wave dissipation in the artificial damping zone. The wave elevations at different locations in the damping zone were also compared in order to appraise the damping capabilities of the respective damping schemes. The optimal numerical damping scheme associated with a proper damping shape over a range of damping zones was proposed for an efficient numerical beach.

A new high-order compact scheme for American options under jump-diffusion processes

Jump-diffusion option pricing models have the ability to fit various implied volatility patterns observed in market option prices. In the partial differential equations framework, pricing an American put when the underlying follows a jump process requires the solution of a partial integro-differential equation. For this problem, second-order finite difference discretisations have been commonly employed. This work develops a new scheme which is based on a high-order compact discretisation of the spatial terms of the equation and a fourth-order time integration scheme. We demonstrate that the scheme is highly accurate for at-the-money American options and oscillation-free greeks are computed.

Numerical Investigations of Reignition in Vortex-Perturbed n-Heptane Nonpremixed Flames

Numerical investigations of reignition in n-heptane nonpremixed flames perturbed by counter-rotating vortex pairs are performed. The conditions simulated are representative of high-pressure combustion chambers with high-Reynolds-number jet flames. A two-dimensional numerical formulation that employs sixth-order spatial accuracy and fourth-order time integration is used to solve the governing equations for compressible, viscous, and reacting flows. The oxidation of n-heptane is modeled using a 34-species mechanism involving 56 reaction steps. Transport is modeled using the unity Lewis number and the mixture-averaged models. For the conditions simulated, reignition occurs through lateral diffusion of species and heat from the adjoining edge flames to the quenched regions and hence cannot be predicted by one-dimensional flamelet models. Following reconnection of the diffusion flame, a vortical pocket travels on the fuel side and acts as a major source of combustion products such as CO and H 2 . The results show that whereas an increase in the vortex velocity scale results in higher rates of reignition and product formation, an increase in the length scale decreases the rate of reignition and increases the amount of products formed in the vortical pocket. The choice of the transport formulation has a minor influence on the predicted flame temperatures and major species concentrations.

Modeling fully nonlinear shallow-water waves and their interactions with cylindrical structures

Introduction of a time-accurate stabilized finite-element approximation for the numerical investigation of fully nonlinear shallow-water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the triangular elements by the Galerkin method, the fourth-order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The Streamline-Upwind Petrov–Galerkin (SUPG) method with cross-wind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Numerical results obtained for cases representing propagation of solitary waves, collisions of two solitary waves, and wave-structure interactions show fairly good agreement with experimental measurements and other published numerical solutions. The comparisons between results from present fully nonlinear wave model and weekly nonlinear wave model are presented and discussed.

A 2-D DNS investigation of extinction and reignition dynamics in nonpremixed flame–vortex interactions

Two-dimensional (2-D) DNS investigations of extinction and reignition dynamics during interactions of laminar nonpremixed flames with counterrotating vortex pairs are performed. The length and velocity scales chosen for the vortices are representative of those in the near fields of high-Reynolds-number jets such as those occurring in Diesel engines. The governing equations are solved with sixth-order spatial discretization and fourth-order time integration. Chemistry is modeled as an irreversible single-step reaction. Local extinction along the symmetry axis, followed by reignition, is observed. The extinction is characterized by strong unsteady effects, which are captured well by 1-D transient diffusion flamelet libraries, provided the time-history of the instantaneous scalar dissipation rate is taken into account. On the other hand, reignition is essentially a 2-D phenomenon involving flame–flame interactions, which are favored for smaller vortices and increasing flame curvature. The effects of unsteadiness and curvature on extinction and reignition are carefully assessed through parametric studies involving a range of vortex and flame characteristics. The interaction outcomes are summarized on Reynolds–Damköhler number (Re–Da) diagrams, which show the combined effects of unsteadiness and curvature on extinction and reignition. The implications of the observed interaction outcomes for turbulent combustion modeling in the near fields of jet diffusion flames are discussed.

Fully nonlinear wave-body interactions with surface-piercing bodies

Fully nonlinear wave-body interactions for stationary surface-piercing single and double bodies are studied by a potential-theory-based fully nonlinear 2D numerical wave tank (NWT). The NWT was developed in time domain by using boundary element method (BEM) with constant panels. MEL free surface treatment and Runge–Kutta fourth-order time integration with smoothing scheme was used for free-surface time simulation. The acceleration-potential scheme is employed to obtain accurate time derivative of velocity potential. Using the steady part of nonlinear force time histories, mean and a series of higher-harmonic force components are calculated and compared with the experimental and numerical results of other researchers. The slow-decaying second-harmonic vertical forces are investigated with particle velocities and corresponding body pressure. Typical patterns of two-body interactions, shielding effect, and the pumping/sloshing modes of water column in various gap distances are investigated. The pumping mode in low frequencies is demonstrated by the comparison of velocity magnitudes.

Extended one-step time-integration schemes for convection-diffusion equations

We first describe a one-parameter family of unconditionally stable third-order time-integration schemes for the convection-diffusion equation: u t + cu x = vu xx , based on the extended trapezoidal formulas of Usmani and Agarwal [1]. Interestingly, there exists a method that is fourth order as well as unconditionally stable. We then describe a one-parameter family of unconditionally stable fourth-order time-integration schemes, based on the extended Simpson rules of Chawla et al. [2]. Again, there exists a method that is fifth order as well as unconditionally stable. The stability and accuracy of the obtained methods are tested, and compared with the widely used method of Crank-Nicolson, by considering three problems of practical interest.