Standard Explicit Methods Research Articles

Multi-Level Solution Strategies for Implicit Time Discontinuous Galerkin Discretizations

The discontinuous Galerkin method and related flavors of high-order spectral element methods provide many well-known benefits for the spatial discretization of partial differential equations such as the Navier–Stokes equations. However, practical problems of engineering relevance such as large-eddy simulation of turbulent flows over complex geometries are computationally intractable by standard explicit time integration methods, necessitating the use of implicit methods. The efficient solution of the nonlinear algebraic systems arising from implicit time integration methods applied to DG discretizations of nonlinear PDEs is challenging; standard linearization methods result in very stiff block-sparse systems with prohibitive computational and memory requirements. This paper presents a low-memory, computationally efficient, implicit solution method that combines a framework of nonlinear polynomial multigrid, adaptive explicit Runge–Kutta smoothers, implicit Jacobian-free coarse level smoothers, nonlinear Krylov subspace acceleration and adaptive time stepping using feedback control of the nonlinear solver convergence rate.

Analytical and Numerical Results for the Diffusion-Reaction Equation When the Reaction Coefficient Depends on Simultaneously the Space and Time Coordinates

We utilize the travelling-wave Ansatz to obtain novel analytical solutions to the linear diffusion–reaction equation. The reaction term is a function of time and space simultaneously, firstly in a Lorentzian form and secondly in a cosine travelling-wave form. The new solutions contain the Heun functions in the first case and the Mathieu functions for the second case, and therefore are highly nontrivial. We use these solutions to test some non-conventional explicit and stable numerical methods against the standard explicit and implicit methods, where in the latter case the algebraic equation system is solved by the preconditioned conjugate gradient and the GMRES solvers. After this verification, we also calculate the transient temperature of a 2D surface subjected to the cooling effect of the wind, which is a function of space and time again. We obtain that the explicit stable methods can reach the accuracy of the implicit solvers in orders of magnitude shorter time.

Numerical evolution of the resistive relativistic magnetohydrodynamic equations: A minimally implicit Runge-Kutta scheme

We present the Minimally-Implicit Runge-Kutta (MIRK) methods for the numerical evolution of the resistive relativistic magnetohydrodynamic (RRMHD) equations, following the approach proposed by Komissarov (2007) of an augmented system of evolution equations to numerically deal with constraints. Previous approaches rely on Implicit-Explicit (IMEX) Runge-Kutta schemes; in general, compared to explicit schemes, IMEX methods need to apply the recovery (which can be very expensive computationally) of the primitive variables from the conserved ones in numerous additional times. Moreover, the use of an iterative process for the recovery could have potential convergence problems, increased by the additional number of required loops. In addition, the computational cost of the previous IMEX approach in comparison with the standard explicit methods is much higher. The MIRK methods are able to deal with stiff terms producing stable numerical evolutions, minimize the number of recoveries needed in comparison with IMEX methods, their computational cost is similar to the standard explicit methods and can actually be easily implemented in numerical codes which previously used explicit schemes. Two standard numerical tests are shown in the manuscript.

Simulation of non-linear structural elastodynamic and impact problems using minimum energy high-order bases

We present the application of simultaneous diagonalization and minimum energy (SDME) high-order finite element modal bases for simulation of transient non-linear elastodynamic problem, including impact cases with Hookean and neo-Hookean hyperelastic materials. The bases are constructed using procedures for simultaneous diagonalization of the internal modes and Schur complement of the boundary modes from the standard nodal and modal bases, constructed using Lagrange and Jacobi polynomials, respectively. The implementation of these bases in a high-order finite element code is straightforward, since the procedure is applied only to the one-dimensional expansion bases. Non-linear transient structural problems with large deformation , hyperelastic materials and impact are solved using the obtained bases with explicit and implicit time integration procedures. Iterative solutions based on preconditioned conjugate gradient methods are considered. The performance of the proposed bases in terms of the number of iterations of pre-conditioned conjugate gradient methods and computational time are compared with the standard nodal and modal bases. The SDME bases are accurate and provide better conditioned equations for iterative solutions in general, leading to substantial reduction in processing time over an order of magnitude compared to other modal bases. Our numerical tests obtained speedups up to 41 using the considered bases when compared to the standard modal basis. Results for a two-dimensional impact problem approximated with the SDME bases showed that the implicit Newmark method performed much better in terms of processing time and contact stress error when compared to the standard explicit central difference method with lumped mass matrix . • Application of simultaneous diagonalization procedure and minimum energy highorder bases to non-linear structural and impact transient problems. • Simple modifications of high-order bases improve conditioning of system of equations obtained from the high-order approximation of the considered problems, with remarkable gains in performance of iterative solvers. • Light variation on the number of pre-conditioned gradient iterations in terms of the polynomial order. • Speed-ups over 40 when compared to the standard modal Jacobi high-order bases used in high-order interpolation.

Semi-Implicit Time Integration with Hessian Eigenvalue Corrections for a Larger Time Step in Molecular Dynamics Simulations.

In molecular dynamics simulations, the limited time step size has been a barrier to simulating long-time behaviors. Implicit time integration methods allow markedly larger time steps than the standard explicit time method, although they have major drawbacks such as overheads solving linear systems and instability of Newton iterations. To overcome these issues, we propose a semi-implicit time integration scheme, the semi-implicit Hessian correction (SimHec) scheme, for overdamped Langevin dynamics. The method focuses on the Hessian matrices of bonded and nonbonded interactions, where components with large negative Hessian eigenvalues are cut off in the linear approximation of momentum equations to avoid instability. The narrow band Hessian matrix enables an efficient parallelized linear solution with an overlapping approximation. We tested SimHec for the interdomain fluctuations in adenylate kinase and the powerstroke transition of myosin II using a coarse-grained protein model. SimHec reproduced the same dynamics as the explicit method, although the transition dynamics tended to be accelerated and fluctuations in bonded potentials were slightly reduced. These deviations were corrected using a hybrid method, SimHec-H, which adds explicit time steps after the semi-implicit time step. The proposed scheme allowed us to use time steps 50-200 times larger than those in explicit time integration, which resulted in a speedup factor of 7-30 taking the overhead into account.

On the realization of explicit Runge-Kutta schemes preserving quadratic invariants of dynamical systems

We implement several explicit Runge-Kutta schemes that preserve quadratic invariants of autonomous dynamical systems in Sage. In this paper, we want to present our package ex.sage and the results of our numerical experiments. In the package, the functions rrk_solve, idt_solve and project_1 are constructed for the case when only one given quadratic invariant will be exactly preserved. The function phi_solve_1 allows us to preserve two specified quadratic invariants simultaneously. To solve the equations with respect to parameters determined by the conservation law we use the elimination technique based on Grbner basis implemented in Sage. An elliptic oscillator is used as a test example of the presented package. This dynamical system has two quadratic invariants. Numerical results of the comparing of standard explicit Runge-Kutta method RK(4,4) with rrk_solve are presented. In addition, for the functions rrk_solve and idt_solve, that preserve only one given invariant, we investigated the change of the second quadratic invariant of the elliptic oscillator. In conclusion, the drawbacks of using these schemes are discussed.

Solving nonlinear parabolic PDEs in several dimensions: Parallelized ESERK codes

There is a very large number of very important situations which can be modeled with nonlinear parabolic partial differential equations (PDEs) in several dimensions. In general, these PDEs can be solved by discretizing in the spatial variables and transforming them into huge systems of ordinary differential equations (ODEs), which are very stiff. Therefore, standard explicit methods require a large number of iterations to solve stiff problems. But implicit schemes are computationally very expensive when solving huge systems of nonlinear ODEs. Several families of Extrapolated Stabilized Explicit Runge-Kutta schemes (ESERK) with different order of accuracy (3 to 6) are derived and analyzed in this work. They are explicit methods, with stability regions extended, along the negative real semi-axis, quadratically with respect to the number of stages s, hence they can be considered to solve stiff problems much faster than traditional explicit schemes. Additionally, they allow the adaptation of the step length easily with a very small cost.Two new families of ESERK schemes (ESERK3 and ESERK6) are derived, and analyzed, in this work. Each family has more than 50 new schemes, with up to 84.000 stages in the case of ESERK6. For the first time, we also parallelized all these new variable step length and variable number of stages algorithms (ESERK3, ESERK4, ESERK5, and ESERK6). These parallelized strategies allow to decrease times significantly, as it is discussed and also shown numerically in two problems. Thus, the new codes provide very good results compared to other well-known ODE solvers. Finally, a new strategy is proposed to increase the efficiency of these schemes, and it is discussed the idea of combining ESERK families in one code, because typically, stiff problems have different zones and according to them and the requested tolerance the optimum order of convergence is different.

Super-Time-Stepping Scheme for Option Pricing

We solve the Black - Scholes equation for option pricing numerically using an Explicit finite difference method. To overcome the stability restriction of the explicit scheme for parabolic partial differential equations in the time step size Courant-Friedrichs-Lewy (CFL) condition, we employ a Super Time Stepping (STS) strategy based on modified Chebyshev polynomial. The numerical results show that the STS scheme boasts of large efficiency gains compared to the standard explicit Euler method.

Evaluation of Implicit‐Explicit Additive Runge‐Kutta Integrators for the HOMME‐NH Dynamical Core

AbstractThe nonhydrostatic High‐Order Method Modeling Environment (HOMME‐NH) atmospheric dynamical core supports acoustic waves that propagate significantly faster than the advective wind speed, thus greatly limiting the time step size that can be used with standard explicit time integration methods. Resolving acoustic waves is unnecessary for accurate climate and weather prediction. This numerical stiffness is addressed herein by considering implicit‐explicit additive Runge‐Kutta (ARK IMEX) methods that can treat the acoustic waves in a stable manner without requiring implicit treatment of nonstiff modes. Various ARK IMEX methods are evaluated for their efficiency in producing accurate solutions, ability to take large time step sizes, and sensitivity to grid cell length ratio. Both the gravity wave test and baroclinic instability test from the 2012 Dynamical Core Model Intercomparison Project are used to recommend 5 of the 27 ARK IMEX methods tested for use in HOMME‐NH.

A Class of Two-Derivative Two-Step Runge-Kutta Methods for Non-Stiff ODEs

In this paper, a new class of two-derivative two-step Runge-Kutta (TDTSRK) methods for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equation (ODEs) is considered. The TDTSRK methods are a special case of multi-derivative Runge-Kutta methods proposed by Kastlunger and Wanner (1972). The methods considered herein incorporate only the first and second derivatives terms of ODEs. These methods possess large interval of stability when compared with other existing methods in the literature. The experiments have been performed on standard problems, and comparisons were made with some standard explicit Runge-Kutta methods in the literature.

The relativistic implicit Particle-in-Cell method

Relativistic Particle-in-Cell (PiC) methods are among the most reliable methods for the investigation of plasma phenomena at the particle scale. Standard explicit and semi-implicit PiC methods are affected by numerical instabilities that restrain the range of admissible simulation parameters, and prevent their application to large domains over long time scales. Here, we present a three-dimensional, fully-implicit algorithm for relativistic PiC simulations that conserves energy exactly (to machine precision) and eliminates numerical instabilities, allowing long-term calculations. We describe the full numerical solution procedure and test the algorithm with several one-and two-dimensional case-studies of kinetic instabilities, comparing the results with theoretical predictions.

Efficient numerical schemes for Chan-Vese active contour models in image segmentation

In this paper, we introduce multi-symplectic Lagrangian variational integrators for solving Chan-Vese active contour models in image segmentation. Energy functionals are discretized firstly, and numerical schemes are derived from discrete Euler-Lagrange equations based on discrete variational principle. Lagrangian variational integrators preserve native differential structure-multi-symplecticity, that makes the numerical methods have a satisfied behavior. Experiments are performed on the benchmark images from literature. We further evaluated the methods in a segmentation database containing 1023 images. It shows that the proposed numerical schemes attain relatively faster convergence rates and better segmentation accuracy. Comparisons with the standard explicit Euler method of the original Chan-Vese model and other fast numerical optimization methods show that the proposed methods have better stability, higher accuracy, and are more robust when dealing with a large number of pictures. This study provides an example for further research to improve the performance of other existing image segmentation methods based on active contour models.

Factorized Runge–Kutta–Chebyshev Methods

The second-order extended stability Factorized Runge–Kutta–Chebyshev (FRKC2) explicit schemes for the integration of large systems of PDEs with diffusive terms are presented. The schemes are simple to implement through ordered sequences of forward Euler steps with complex stepsizes, and easily parallelised for large scale problems on distributed architectures. Preserving 7 digits for accuracy at 16 digit precision, the schemes are theoretically capable of maintaining internal stability for acceleration factors in excess of 6000 with respect to standard explicit Runge-Kutta methods. The extent of the stability domain is approximately the same as that of RKC schemes, and a third longer than in the case of RKL2 schemes. Extension of FRKC methods to fourth-order, by both complex splitting and Butcher composition techniques, is also discussed.A publicly available implementation of FRKC2 schemes may be obtained from maths.dit.ie/frkc

Fully implicit finite element method for the modeling of free surface flows with surface tension effect

A new approach based on the use of the Newton and level set methods allows to follow the motion of interfaces with surface tension immersed in an incompressible Newtonian fluid. Our method features the use of a high-order fully implicit time integration scheme that circumvents the stability issues related to the explicit discretization of the capillary force when capillary effects dominate. A strategy based on a consistent Newton–Raphson linearization is introduced, and performances are enhanced by using an exact Newton variant that guarantees a third-order convergence behavior without requiring second-order derivatives. The problem is approximated by mixed finite elements, while the anisotropic adaptive mesh refinements enable us to increase the computational accuracy. Numerical investigations of the convergence properties and comparisons with benchmark results provide evidence regarding the efficacy of the methodology. The robustness of the method is tested with respect to the standard explicit method, and stability is maintained for significantly larger time steps compared with those allowed by the stability condition. Copyright © 2016 John Wiley & Sons, Ltd.

Large Time Step Roe scheme for a common 1D two-fluid model

We present the Large Time Step (LTS) extension of the Roe scheme and apply it to a standard two-fluid model. Herein, LTS denotes a class of explicit methods that are not limited by the CFL (Courant–Friedrichs–Lewy) condition, allowing us to use very large time steps compared to standard explicit methods. The LTS method was originally developed in the nineteen eighties (LeVeque, 1985), where the Godunov scheme was extended to the LTS Godunov scheme. In the present work, the relaxation of the CFL condition is achieved by increasing the domain of dependence. This might lead to difficulties when it comes to boundary and source terms treatment. We address and discuss these difficulties and propose different ways to treat them. For a shock tube test case, where there are neither source terms nor difficulties associated with the boundaries, the method increases both accuracy and efficiency. For a water faucet test case that includes a source term, the method increases the efficiency, while the accuracy strongly depends on the appropriate treatment of boundary conditions and source terms.

Appropriate use of the particle-in-cell method in low temperature plasmas: Application to the simulation of negative ion extraction

The Particle-In-Cell Monte Carlo Collision (PIC MCC) method has been used by different authors in the last ten years to describe negative ion extraction in the context of neutral beam injection for fusion. Questionable results on the intensity and profile of the extracted negative ion beamlets have been presented in several recently published papers. Using a standard explicit PIC MCC method, we show that these results are due to a non-compliance with the constraints of the numerical method (grid spacing, number of particles per cell) and to a non-physical generation of the simulated plasma. We discuss in detail the conditions of mesh convergence and plasma generation and show that the results can significantly deviate from the correct solution and lead to unphysical features when the constraints inherent to the method are not strictly fulfilled. This paper illustrates the importance of verification in any plasma simulation. Since the results presented in this paper have been obtained with careful verification of the method, we propose them as benchmarks for future comparisons between different simulation codes for negative ion extraction.

Accelerated trinomial trees applied to American basket options and American options under the Bates model

The accelerated trinomial tree (ATT) is a derivatives pricing lattice method that circumvents the restrictive time step condition inherent in standard trinomial trees and explicit finite difference methods (FDMs), in which the time step must scale with the square of the spatial step. ATTs consist of L uniform supersteps, each of which contains an inner lattice/trinomial tree with N nonuniform subtime steps. Similarly to implicit FDMs, the size of the superstep in ATTs, a function of N, is constrained primarily by accuracy demands. ATTs can price options up to N times faster than standard trinomial trees (explicit FDMs). ATTs can be interpreted as using risk-neutral extended probabilities: extended in the sense that values can lie outside the range [0, 1] on the substep scale but aggregate to probabilities within the range [0,1] on the superstep scale. Hence, it is only strictly at the end of each superstep that a practically meaningful solution may be extracted from the tree. We demonstrate that ATTs with L supersteps are more efficient or have comparable efficiency to competing implicit methods that use L time steps in pricing Black–Scholes American put options and two-dimensional American basket options. Crucially, this performance is achieved using an algorithm that requires only a modest modification of a standard trinomial tree. This is in contrast to implicit FDMs, which may be relatively complex in their implementation. We also extend ATTs to the pricing of American options under the Heston model and the Bates model in order to demonstrate the general applicability of the approach.

Minimally implicit Runge-Kutta methods for Resistive Relativistic MHD

The Relativistic Resistive Magnetohydrodynamic (RRMHD) equations are a hyperbolic system of partial differential equations used to describe the dynamics of relativistic magnetized fluids with a finite conductivity. Close to the ideal magnetohydrodynamic regime, the source term proportional to the conductivity becomes potentially stiff and cannot be handled with standard explicit time integration methods. We propose a new class of methods to deal with the stiffness fo the system, which we name Minimally Implicit Runge-Kutta methods. These methods avoid the development of numerical instabilities without increasing the computational costs in comparison with explicit methods, need no iterative extra loop in order to recover the primitive (physical) variables, the analytical inversion of the implicit operator is trivial and the several stages can actually be viewed as stages of explicit Runge-Kutta methods with an effective time-step. We test these methods with two different one-dimensional test beds in varied conductivity regimes, and show that our second-order schemes satisfy the theoretical expectations.

Efficient simulations of tubulin-driven axonal growth.

This work concerns efficient and reliable numerical simulations of the dynamic behaviour of a moving-boundary model for tubulin-driven axonal growth. The model is nonlinear and consists of a coupled set of a partial differential equation (PDE) and two ordinary differential equations. The PDE is defined on a computational domain with a moving boundary, which is part of the solution. Numerical simulations based on standard explicit time-stepping methods are too time consuming due to the small time steps required for numerical stability. On the other hand standard implicit schemes are too complex due to the nonlinear equations that needs to be solved in each step. Instead, we propose to use the Peaceman-Rachford splitting scheme combined with temporal and spatial scalings of the model. Simulations based on this scheme have shown to be efficient, accurate, and reliable which makes it possible to evaluate the model, e.g. its dependency on biological and physical model parameters. These evaluations show among other things that the initial axon growth is very fast, that the active transport is the dominant reason over diffusion for the growth velocity, and that the polymerization rate in the growth cone does not affect the final axon length.

Flexible multibody simulation using hybrid integration scheme

One issue in the dynamic simulation of flexible multibody system is poor computation efficiency, which is due to high frequency components in the solution associated with a deformable body. Standard explicit numerical methods should take very small time steps in order to satisfy the absolute stability condition for the high frequency components and, in turn, the computational efficiency deteriorates. In this study, a hybrid integration scheme is applied to solve the equations of motion of a flexible multibody system for achieving better computational efficiency. The computation times and simulation results are compared between the hybrid scheme and conventional methods. The results demonstrate that the efficiency of a flexible multibody simulation can be improved by using the hybrid scheme.