Explicit Runge-Kutta Scheme Research Articles

A novel radial basis neural network for the Zika virus spreading model

The motive of current investigations is to design a novel radial basis neural network stochastic structure to present the numerical representations of the Zika virus spreading model (ZVSM). The mathematical ZVSM is categorized into humans and vectors based on the susceptible S(q), exposed E(q), infected I(q) and recovered R(q), i.e., SEIR. The stochastic performances are designed using the radial basis activation function, feed forward neural network, twenty-two numbers of neurons along with the optimization of Bayesian regularization in order to solve the ZVSM. A dataset is achieved using the explicit Runge-Kutta scheme, which is used to reduce the mean square error (MSE) based on the process of training for solving the nonlinear ZVSM. The division of the data is categorized into training, which is taken as 78 %, while 11 % for both authentication and testing. Three different cases of the nonlinear ZVSM have been taken, while the scheme’s correctness is performed through the matching of the results. Furthermore, the reliability of the scheme is observed by applying different performances of regression, MSE, error histograms and state transition.

On the Selection of Weights for Difference Schemes to Approximate Systems of Differential Equations

We consider the problem of determining the weights of difference schemes whose form is specified by a particular symbolic expression. The order of approximation of the differential equation is equal to a given number. To solve it, it was propose to proceed from considering systems of differential equations of a general form to one scalar equation. This method provides us with some values for the weights, which we propose to test using Richardson’s method. The method was shown to work in the case of low-order schemes. However, when transitioning from the scalar problem to the vector and nonlinear problems, the reduction of the order of the scheme, whose weights are selected for the scalar problem, occurs in different families of schemes. This was first discovered when studying the Shanks scheme, which belongs to the family of explicit Runge–Kutta schemes. This does not deteriorate the proposed strategy itself concerning the simplification of the weight-determination problem, which should include a clause on mandatory testing of the order using the Richardson method.

A high-order finite difference method for moving immersed domain boundaries and material interfaces

We present a high-order sharp treatment of immersed moving domain boundaries and material interfaces, and apply it to the advection-diffusion equation in two and three dimensions. The spatial discretization combines dimension-split finite difference schemes with an immersed boundary treatment based on a weighted least-squares reconstruction of the solution, providing stable discretizations with up to sixth order accuracy for diffusion terms and third order accuracy for advection terms. The temporal discretization relies on a novel strategy for maintaining high-order temporal accuracy in problems with moving boundaries that minimizes implementation complexity and allows arbitrary explicit or diagonally-implicit Runge-Kutta schemes. The approach is broadly compatible with popular PDE-specialized Runge-Kutta time integrators, including low-storage, strong stability preserving, and diagonally implicit schemes. Through numerical experiments we demonstrate that the full discretization maintains high-order spatial and temporal accuracy in the presence of complex 3D geometries and for a range of boundary conditions, including Dirichlet, Neumann, and flux conditions with large jumps in coefficients.

A finite-volume framework to solve the Fokker–Planck equation for fiber orientation kinetics

In this work, a new solver, FPSolve, is developed to study fiber orientation kinetics using the Fokker–Planck (FP) equation. The solver employs the finite-volume method. The FP equation is discretized on unstructured cubed-sphere grids using the centered differencing scheme (CDS) or a blend of the CDS and the upwind differencing scheme. Time integration is performed using a second-order two-stage explicit Runge–Kutta scheme. Different shape factors and rotational diffusion coefficients are implemented to study suspensions in dilute to semiconcentrated regimes. The verification of the solver is performed for the fiber orientation in simple shear flow up to a Peclet number of 105. Grid independence analysis is presented to show the second-order accuracy of FPSolve. It is demonstrated that the solver does not need stabilization by upwinding. Simulations for semiconcentrated suspensions are performed using the model of Ferec et al. (2014). Time-accurate solutions of the FP equation with explicit time stepping for this model are presented for the first time.

Abaqus implementation of a large family of finite viscoelasticity models

In this paper, we introduce an Abaqus UMAT subroutine for a family of constitutive models for the viscoelastic response of isotropic elastomers of any compressibility – including fully incompressible elastomers – undergoing finite deformations. The models can be chosen to account for a wide range of non-Gaussian elasticities, as well as for a wide range of nonlinear viscosities. From a mathematical point of view, the structure of the models is such that the viscous dissipation is characterized by an internal variable Cv, subject to the physically-based constraint detCv=1, that is solution of a nonlinear first-order ODE in time. This ODE is solved by means of an explicit Runge–Kutta scheme of high order capable of preserving the constraint detCv=1 identically. The accuracy and convergence of the code is demonstrated numerically by comparison with an exact solution for several of the Abaqus built-in hybrid finite elements, including the simplicial elements C3D4H and C3D10H and the hexahedral elements C3D8H and C3D20H. The last part of this paper is devoted to showcasing the capabilities of the code by deploying it to compute the homogenized response of a bicontinuous rubber blend.

Tetra geoemtric mean Runge-Kutta analysis of bioeconomic fisheries model

The optimal management and utilization of natural resources is eventually important, due to increase in the consumption of natural resources and in view of this, the main aim of fisheries is to maintain the marine organisms at levels that produces maximum sustainable yield (MSY). In this study, the one step numerical approximations schemes are developed to find the solution of such bioeconomic fisheries models whose analytical solution is not possible. In the proposed work, the explicit Runge-Kutta scheme based on fourth order tetra geometric mean is established and this method solve fishery management models of Schaefer and Gordon-Schaefer. The models analyze bioeconomic growth rate, carrying capacity, total and marginal costs and revenues. In solving, the models through proposed tetra geometric mean Runge-Kutta method, it is obtained that the method is easy to implement. The results obtained are much better as compared to other methods when compared in terms of errors and central processing unit (CPU) time.

General-relativistic Radiation Transport Scheme in Gmunu. I. Implementation of Two-moment-based Multifrequency Radiative Transfer and Code Tests

We present the implementation of a two-moment-based general-relativistic multigroup radiation transport module in the General-relativistic multigrid numerical (Gmunu) code. On top of solving the general-relativistic magnetohydrodynamics and the Einstein equations with conformally flat approximations, the code solves the evolution equations of the zeroth- and first-order moments of the radiations in the Eulerian-frame. An analytic closure relation is used to obtain the higher order moments and close the system. The finite-volume discretization has been adopted for the radiation moments. The advection in spatial space and frequency-space are handled explicitly. In addition, the radiation–matter interaction terms, which are very stiff in the optically thick region, are solved implicitly. The implicit–explicit Runge–Kutta schemes are adopted for time integration. We test the implementation with a number of numerical benchmarks from frequency-integrated to frequency-dependent cases. Furthermore, we also illustrate the astrophysical applications in hot neutron star and core-collapse supernovae modelings, and compare with other neutrino transport codes.

Neural network representation of time integrators

AbstractDeep neural network (DNN) architectures are constructed that are the exact equivalent of explicit Runge–Kutta schemes for numerical time integration. The network weights and biases are given, that is, no training is needed. In this way, the only task left for physics‐based integrators is the DNN approximation of the right‐hand side. This allows to clearly delineate the approximation estimates for right‐hand side errors and time integration errors. The architecture required for the integration of a simple mass‐damper‐stiffness case is included as an example.

Singly TASE Operators for the Numerical Solution of Stiff Differential Equations by Explicit Runge–Kutta Schemes

In this paper new explicit integrators for numerical solution of stiff evolution equations are proposed. As shown by Bassenne, Fu and Mani in (J Comput Phys 424:109847, 2021), the action on the original vector field of the stiff equations of an appropriate time-accurate and highly-stable explicit (TASE) linear operator, allows us to use explicit Runge–Kutta (RK) schemes with these modified equations so that the resulting algorithm becomes stable for the original stiff equations. Here a new family of TASE operators is considered. The new operators, called Singly TASE, have the advantage over the TASE operators of Bassenne et al. that the action on the vector field depends on the powers of the inverse of only one matrix, which can be computationally more simple, without loosing stability properties. A complete study of the linear stability properties of k–stage, kth–order explicit RK schemes under the action of Singly TASE operators of the same order is carried out for k le 4. For orders two, three and four, particular schemes that are nearly strongly A–stable and therefore suitable for stiff problems are devised. Further, explicit RK schemes with orders three and four that can be implemented with only two storage locations under the action of Singly TASE operators of the same order are discussed. A particular implementation of the classical four–stage fourth–order RK scheme with two Singly TASE operators is presented. A set of numerical experiments has been conducted to demonstrate the performance of the new schemes by comparing with previous RKTASE and other established methods. The main conclusion is that the new integrators provide a very simple solver for stiff systems with good stability properties and avoids the difficulties of using implicit algorithms.

Local wall heating effects on aeroacoustic field radiated by an isolated square cylinder in a laminar flow

The aim of this work is to gain an insight on the effect of the wall heating on the aeroacoustic sound radiated by bluff bodies in laminar flows. In particular, the local thermal treatment of the wall boundary was investigated as a possible method for active controlling the emitted noise. This technique was studied performing direct numerical simulations of the aeroacoustic noise produced by an isolated square cylinder operating at a Reynolds and Mach numbers equal to 150 and 0.2, respectively. In the considered case, the Karman vortex street deriving by the flow/cylinder interaction, produces a lift and drag pulsation on the body surface, leading to a dipolar-like noise emission. In this context, different local thermal fluxes were applied to the cylinder wall in order to reduce its aerodynamic forces fluctuation and, consequently, the associated pressure disturbance that produces the radiated sound. The computations are performed using an OpenFOAM solver that adopts an explicit Runge-Kutta scheme for time integration and a second-order, energy conserving scheme for the convective part of the Eulerian flux. Moreover, the spurious numerical waves reflections at the far-field boundary are damped adopting a sponge-layer approach.

Embedded paired explicit Runge-Kutta schemes

Recently, Paired Explicit Runge-Kutta (P-ERK) schemes were introduced to accelerate the solution of locally-stiff systems of equations. P-ERK schemes use different numbers of active stages based on local stiffness criteria, significantly reducing computational cost relative to classical explicit Runge-Kutta schemes. In the current work we present a framework for incorporating an embedded pair within the original P-ERK formulation. By design, the embedded scheme has a slightly smaller stability limit than its corresponding base scheme. When combined with a suitable controller, the difference between the base and embedded solution approximations can be used to adapt the time step size to be as close as possible to, without exceeding, the stability limit of the base scheme. Numerical results using the high-order flux reconstruction approach for spatial discretization of the compressible Navier-Stokes equations are presented for flow over a circular cylinder, a plunging and pitching NACA 0012 airfoil, and a stalled NACA 0020 airfoil. It is demonstrated that adaptive time stepping using embedded P-ERK schemes yields excellent agreement with reference data, while being up to seven times less computationally expensive than classical embedded pairs for all cases.

On a generalization of time-accurate and highly-stable explicit operators for stiff problems

In this work we propose a generalization of the family of Time-Accurate and highly-Stable Explicit (TASE) operators recently introduced by Calvo, Montijano and Rández (2021). In this family the TASE operator of order p depends on p free real parameters. Here we consider operators that can also be defined by complex parameters occurring in conjugate pairs. Despite this choice the calculations continue to be done only in real arithmetic, thus not burdening the computational cost of the previous version of the family. Conversely, this generalization leads to improve both the accuracy and stability properties of explicit Runge-Kutta schemes supplemented with TASE operators. Numerical experiments showing the competitiveness of the methods proposed in this paper with respect to classical integrators for stiff problems are also presented.

The numerical error of the Xinanjiang model

The numerical error of the Xinanjiang (XAJ) model has long been ignored, despite its wide application in China. Separating the mathematically formulated model from the computationally formulated model and further solving by an appropriate time-stepping scheme provides a chance to systematically examine the numerical error of the XAJ model because the exact analytical solution is difficult to obtain owing to its nonlinearity. The mathematically formulated model is proposed by deriving a state-space representation of the XAJ model by identifying the state variables, fluxes, and governing differential equations. Five experiments were conducted around the original XAJ model and its state-space representation with different parameter groups, input series, and time-stepping schemes to examine the numerical error at the submodule and model levels. We found that only the tension water storage capacity curve submodule of the original XAJ model is numerical error-free, and the adaptive-step explicit fourth-order Runge-Kutta scheme is the most efficient because it balances the accuracy, efficiency, and physical constraint preservation. The numerical error of the total amount of the discharge at the watershed outlet within a time period (Q∗) produced by the original XAJ model is relatively large, which mainly comes from the free water storage capacity curve submodule. The large numerical error of Q∗ occurs with high precipitation intensity, and decreases as the flood receded. The submodules related to the runoff generation process introduce and amplify numerical errors, while the submodules related to the runoff concentration process has the ability to alleviate numerical errors. This study helps to deepen our understanding of the numerical error of the XAJ model, and thus guides its application and modification.

Implementation of low-storage Runge-Kutta time integration schemes in scalable asynchronous partial differential equation solvers

The asynchronous computing method based on finite-difference schemes has shown promise in significantly improving the scalability of time-dependent partial differential equation (PDE) solvers by either relaxing data synchronization or avoiding communication between processing elements (PEs) on massively parallel machines. This method uses high-order accurate asynchrony-tolerant (AT) schemes coupled with appropriate time integration schemes to provide the desired accuracy required by high-fidelity solvers such as the direct numerical simulations for fluid flow. For time integration, Runge-Kutta (RK) schemes, particularly the low-storage implementation, are widely used due to their ability to provide good stability properties and be computationally efficient. However, the implementation of AT schemes with multi-stage RK schemes necessitates an over-decomposition of the spatial domain in a parallel setting, leading to increased message sizes for communication and redundant computations. In this paper, we propose a novel method to couple asynchrony-tolerant and low-storage explicit RK schemes in solving time-dependent PDEs that would result in a significant reduction in communications and relaxed synchronizations. We develop new asynchrony-tolerant schemes for ghost or buffer point updates that are necessary to maintain desired order of accuracy. The accuracy of this method is investigated, both theoretically and numerically, using simple one-dimensional linear model equations. Thereafter, we demonstrate the scalability of the proposed numerical method through three-dimensional simulations of decaying Burgers' turbulence, performed using two different asynchronous algorithms: communication-avoiding and synchronization-avoiding algorithms. The scalability studies up to 27,000 cores were found to yield speed-ups up to 6× compared to a baseline synchronous algorithm. Overall, the proposed approach shows the potential to improve the scalability of exascale PDE solvers significantly.

Conservative finite difference schemes for dynamical systems

The article presents the implementation of one of the approaches to the integration of dynamical systems, which preserves algebraic integrals in the original fdm for Sage system. This approach, which goes back to the paper by del Buono and Mastroserio, makes it possible, based on any two explicit difference schemes, including any two explicit Runge-Kutta schemes, to construct a new numerical algorithm for integrating a dynamical system that preserves the given integral. This approach has been implemented and tested in the original fdm for Sage system. Details and implementation difficulties are discussed. For testing, two Runge-Kutta schemes were taken having the same order, but different Butcher tables, which does not complicate the method due to paralleling. Two examples are considered - a linear oscillator and a Jacobi oscillator with two quadratic integrals. The second example shows that the preservation of one integral of motion does not lead to the conservation of the other. Moreover, this method allows us to propose a practical application of the well-known ambiguity in the definition of Butcher tables.

An adaptive nonhydrostatic dynamical core using a multimoment finite‐volume method on a cubed sphere

Abstract: An adaptive nonhydrostatic atmospheric dynamical core was developed on a Cartesian grid and extended to spherical geometry with the application of a cubed‐sphere grid. To assure a practical Courant‐Friedrichs‐Lewy number for stability, a horizontally explicit and vertically implicit algorithm was applied in this model through a third‐order implicit–explicit Runge–Kutta scheme. A two‐dimensional adaptive grid was applied in the horizontal directions to generate the computational meshes, with variable resolutions according to the evolution of the predicted variables during the simulations, while the vertical grid is fixed considering the characteristics of atmospheric flows and the computational efficiency in practical applications. The Berger–Oliger block‐structured adaptive algorithm was adopted in this study. Blocks with different resolutions can be constructed straightforwardly on each patch of the cubed sphere and an algorithm to exchange both solution and block information between adjacent patches was designed in order to implement a global model. The nonhydrostatic governing equations are solved by a three‐point multimoment constrained finite‐volume scheme. Using the compact stencil for spatial reconstructions, the interpolation operations between coarse–fine blocks can be implemented efficiently and the local‐based scheme is also helpful to suppress the computational modes around the coarse–fine interfaces due to the abrupt change of grid resolution. Additionally, flux corrections were conducted along the block boundaries and the resulting dynamical core is rigorously conservative. The proposed model was evaluated by calculating several idealized benchmark tests, and the effectiveness of the adaptive model in saving computational costs was verified in this study.

A High-Order Residual-Based Viscosity Finite Element Method for the Ideal MHD Equations

We present a high order, robust, and stable shock-capturing technique for finite element approximations of ideal MHD. The method uses continuous Lagrange polynomials in space and explicit Runge-Kutta schemes in time. The shock-capturing term is based on the residual of MHD which tracks the shock and discontinuity positions, and adds sufficient amount of viscosity to stabilize them. The method is tested up to third order polynomial spaces and an expected fourth-order convergence rate is obtained for smooth problems. Several discontinuous benchmarks such as Orszag-Tang, MHD rotor, Brio-Wu problems are solved in one, two, and three spacial dimensions. Sharp shocks and discontinuity resolutions are obtained.

An Extension of Gmunu: General-relativistic Resistive Magnetohydrodynamics Based on Staggered-meshed Constrained Transport with Elliptic Cleaning

We present the implementation of general-relativistic resistive magnetohydrodynamics solvers and three divergence-free handling approaches adopted in the General-relativistic multigrid numerical (Gmunu) code. In particular, implicit–explicit Runge–Kutta schemes are used to deal with the stiff terms in the evolution equations for small resistivity. The three divergence-free handling methods are (i) hyperbolic divergence cleaning (also known as the generalized Lagrange multiplier), (ii) staggered-meshed constrained transport schemes, and (iii) elliptic cleaning through a multigrid solver, which is applicable in both cell-centered and face-centered (stagger grid) magnetic fields. The implementation has been tested with a number of numerical benchmarks from special-relativistic to general-relativistic cases. We demonstrate that our code can robustly recover from the ideal magnetohydrodynamics limit to a highly resistive limit. We also illustrate the applications in modeling magnetized neutron stars, and compare how different divergence-free handling methods affect the evolution of the stars. Furthermore, we show that the preservation of the divergence-free condition of the magnetic field when using staggered-meshed constrained transport schemes can be significantly improved by applying elliptic cleaning.

Third-order Paired Explicit Runge-Kutta schemes for stiff systems of equations

The ability to advance locally-stiff systems of equations in time depends on accurate and efficient temporal schemes. Recently, a new family of Paired Explicit Runge-Kutta (P-ERK) methods has been proposed. This approach allows different Runge-Kutta schemes with different numbers of active stages to be assigned based on local stiffness criteria. Whereas the original P-ERK formulation was only second-order accurate, in this paper we propose a new third-order family. We present the general formulation for these schemes, and then optimize them for the high-order discontinuous Galerkin method recovered via the flux reconstruction approach. We then verify that these schemes obtain their designed third-order accuracy for non-linear systems of equations. Performance results show that they achieve speedup factors up to four relative to a classical third-order Runge-Kutta methods for laminar and turbulent flow over an SD7003 airfoil, and turbulent flow over a tandem sphere configuration. Based on these results, this new family of third-order P-ERK schemes provides an appealing approach for accurate and efficient solution of locally-stiff systems of equations.

A nearly-conservative, high-order, forward Lagrange–Galerkin method for the resolution of compressible flows on unstructured triangular meshes

In this work, we present a novel Lagrange–Galerkin method for the resolution of compressible and inviscid flows. The scheme considers: (i) high-order continuous space discretizations on unstructured triangular meshes, (ii) high-order implicit–explicit Runge-Kutta schemes for the time discretization, (iii) conservation of mass, momentum and total energy, as long as some integrals in the formulation are computed exactly, and (iv) subgrid-stabilization and discontinuity-capturing operators based on Brenner's model [51] (2006) for viscous flows. The method has been tested on several benchmark problems using a fourth-order time-marching formula and up to fifth-order continuous finite elements, yielding the expected results both for smooth and discontinuous solutions.