Time Step Restriction Research Articles

Implicit–explicit schemes for nonlinear nonlocal equations with a gradient flow structure in one space dimension

Nonlinear convection–diffusion equations with nonlocal flux and possibly degenerate diffusion arise in various contexts including interacting gases, porous media flows, and collective behavior in biology. Their numerical solution by an explicit finite difference method is costly due to the necessity of discretizing a local spatial convolution for each evaluation of the convective numerical flux, and due to the disadvantageous Courant–Friedrichs–Lewy (CFL) condition incurred by the diffusion term. Based on explicit schemes for such models devised in the study of Carrillo et al. a second‐order implicit–explicit Runge–Kutta (IMEX‐RK) method can be formulated. This method avoids the restrictive time step limitation of explicit schemes since the diffusion term is handled implicitly, but entails the necessity to solve nonlinear algebraic systems in every time step. It is proven that this method is well defined. Numerical experiments illustrate that for fine discretizations it is more efficient in terms of reduction of error versus central processing unit time than the original explicit method. One of the test cases is given by a strongly degenerate parabolic, nonlocal equation modeling aggregation in study of Betancourt et al. This model can be transformed to a local partial differential equation that can be solved numerically easily to generate a reference solution for the IMEX‐RK method, but is limited to one space dimension.

A Novel Efficient WLP-Based BOR FDTD Method With Explicit Treating Ideology

In this paper, we proposed a novel efficient weighted Laguerre polynomial (WLP)-based finite-difference time-domain (FDTD) method with explicit treating ideology, which is extremely useful for problems with very fine structures in the body of revolution (BOR) system. By combining the explicit treating ideology and WLP technology, the limitation of time step $\Delta \text{t}$ is effectively eliminated, which results in greatly improved performance. The performing idea of the proposed method can be roughly divided into two parts. First, the conventional WLP-based BOR FDTD method is flexibly transformed into a new one with certain direction explicit calculation by matrix transformation. Second, the initial value calculation and iteration calculation which are based on the different perturbation term are introduced to improve the convergence speed, the efficiency and accuracy of the proposed method. Meanwhile, the proofing results show that the convergence condition of the proposed method is relaxed. The stability analysis shows that the stability condition is determined by the smaller one of the spatial increments $\Delta \rho $ and $\Delta \text{z}$ . Finally, two scattering numerical examples are given to demonstrate the computational accuracy and efficiency of the proposed method.

Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces

We analyze two types of summation-by-parts finite difference operators for approximating the second derivative with variable coefficient. The first type uses ghost points, while the second type does not use any ghost points. A previously unexplored relation between the two types of summation-by-parts operators is investigated. By combining them we develop a new fourth order accurate finite difference discretization with hanging nodes on the mesh refinement interface. We take the model problem as the two-dimensional acoustic wave equation in second order form in terms of acoustic pressure, and prove energy stability for the proposed method. Compared to previous approaches using ghost points, the proposed method leads to a smaller system of linear equations that needs to be solved for the ghost point values. Another attractive feature of the proposed method is that the explicit time step does not need to be reduced relative to the corresponding periodic problem. Numerical experiments, both for smoothly varying and discontinuous material properties, demonstrate that the proposed method converges to fourth order accuracy. A detailed comparison of the accuracy and the time-step restriction with the simultaneous-approximation-term penalty method is also presented.

On the Variable Two-Step IMEX BDF Method for Parabolic Integro-differential Equations with Nonsmooth Initial Data Arising in Finance

In this paper the implicit-explicit (IMEX) two-step backward differentiation formula (BDF2) method with variable step-size, due to the nonsmoothness of the initial data, is developed for solving parabolic partial integro-differential equations (PIDEs), which describe the jump-diffusion option pricing model in finance. It is shown that the variable step-size IMEX BDF2 method is stable for abstract PIDEs under suitable time step restrictions. Based on the time regularity analysis of abstract PIDEs, the consistency error and the global error bounds for the variable step-size IMEX BDF2 method are provided. After time semidiscretization, spatial differential operators are treated by using finite difference methods, and the jump integral is computed using the composite trapezoidal rule. A local mesh refinement strategy is also considered near the strike price because of the nonsmoothness of the payoff function. Numerical results illustrate the effectiveness of the proposed method for European and American options under jump-diffusion models.

A High Order Positivity Preserving DG Method for Coagulation-Fragmentation Equations

We design, analyze, and numerically validate a novel discontinuous Galerkin (DG) method for solving the coagulation-fragmentation equations. The DG discretization is applied to the conservative form of the model, with flux terms evaluated by Gaussian quadrature with $Q=k+1$ quadrature points for polynomials of degree $k$. The first moment (total mass) is naturally conserved by the scheme construction, and the positivity of the mass density is enforced by the use of a scaling limiter based on positive cell averages. The positivity of cell averages is shown to propagate by the time discretization, provided a proper time step restriction is imposed.

Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws

In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh2 and the second order TVD-RK scheme needs $ \tau \le \rho {h}^{\frac{4}{3}}$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h.

A Moment Limiter for the Discontinuous Galerkin Method on Unstructured Triangular Meshes

We propose a second order limiter for the discontinuous Galerkin method applied to hyperbolic conservation laws. We prove that under a suitable time step restriction, the limiter enforces the local...

Positivity Preserving Limiters for Time-Implicit Higher Order Accurate Discontinuous Galerkin Discretizations

Currently, nearly all positivity preserving discontinuous Galerkin (DG) discretizations of partial differential equations are coupled with explicit time integration methods. Unfortunately, for many problems this can result in severe time-step restrictions. The techniques used to develop explicit positivity preserving DG discretizations cannot, however, easily be combined with implicit time integration methods. In this paper, we therefore present a new approach. Using Lagrange multipliers, the conditions imposed by the positivity preserving limiters are directly coupled to a DG discretization combined with a diagonally implicit Runge--Kutta time integration method. The positivity preserving DG discretization is then reformulated as a Karush--Kuhn--Tucker (KKT) problem, which is frequently encountered in constrained optimization. Since the limiter is only active in areas where positivity must be enforced, it does not affect the higher order DG discretization elsewhere. The resulting nonsmooth nonlinear algebraic equations have, however, a different structure compared to most constrained optimization problems. We therefore develop an efficient active set semismooth Newton method that is suitable for the KKT formulation of time-implicit positivity preserving DG discretizations. Convergence of this semismooth Newton method is proven using a specially designed quasi-directional derivative of the time-implicit positivity preserving DG discretization. The time-implicit positivity preserving DG discretization is demonstrated for several nonlinear scalar conservation laws, which include the advection, Burgers, Allen--Cahn, Barenblatt, and Buckley--Leverett equations.

A conservative penalisation strategy for the semi-implicit time discretisation of the incompressible elastodynamics equation

The principal aim of this work is to provide an adapted numerical scheme for the approximation of elastic wave propagation in incompressible solids. We rely on high-order conforming finite element with mass lumping for space discretisation and implicit/explicit, second-order, energy-preserving time discretisation. The time step restriction only depends on the shear wave velocity and at each time step a Poisson problem must be solved to account for the incompressibility constraint that is imposed by penalisation techniques.

Fast time‐explicit micro‐heterogeneous wave propagation

AbstractUsing explicit time stepping schemes in the context of wave propagation is very popular due to simplicity and fast computations. However, stability can only be guaranteed if a so‐called CFL condition holds that bounds the time step size with respect to the spatial mesh size. This is a severe restriction in case of micro‐heterogeneous media where spatial finescale features need to be resolved. We show how numerical homogenization in space both reduces the number of degrees of freedom in space and weakens the time step restriction.

Partitioned Time Stepping Method for a Dual-Porosity-Stokes Model

In this report, we study a partitioned time stepping algorithm for a dual-porosity-Stokes model, which consists of dual-porosity media and macrofractures/conduits in the coupled system. More specifically, the dual-porosity-Stokes model uses two pressures, the matrix pressure and the fracture pressure, to couple with the Stokes equations. There are four physically valid interface conditions to couple the two models on the interface, including a no-exchange condition, a mass balance condition, a force balance condition, and the Beavers–Joseph condition. To decouple the complex model into three simple sub-problems and solve them separately, we propose a partitioned time stepping method. It solves one, uncoupled matrix pressure equation, microfracture pressure equation and Stokes equation per time step. Under a modest time step restriction of the form $$\triangle t\le C$$ (depending on physical parameters) , we prove the stability of the method. We also derive its optimal error estimates. Numerical tests verify the theoretical results.

Numerical modeling of wildland surface fire propagation by evolving surface curves

We introduce a new approach to wildland fire spread modeling. We evolve a 3-D surface curve, which represents the fire perimeter on the topography, as a projection to a horizontal plane. Our mathematical model is based on the empirical laws of the fire spread influenced by the fuel, wind, terrain slope, and shape of the fire perimeter with respect to the topography (geodesic and normal curvatures). To obtain the numerical solution, we discretize the arising intrinsic partial differential equation by a semi-implicit scheme with respect to the curvature term. For the advection term discretization, we use the so-called inflow-implicit/outflow-explicit approach and an implicit upwind technique which guarantee the solvability of the corresponding linear systems by an efficient tridiagonal solver without any time step restriction and also the robustness with respect to singularities. A fast treatment of topological changes (splitting and merging of the curves) is described and shown on examples as well. We show the experimental order of convergence of the numerical scheme, we demonstrate the influence of the fire spread model parameters on a testing and real topography, and we reconstruct a simulated grassland fire as well.

An unconditional stable compact fourth-order finite difference scheme for three dimensional Allen–Cahn equation

In this paper, we present an unconditional stable linear high-order finite difference scheme for three dimensional Allen–Cahn equation. This scheme, which is based on a backward differentiation scheme combined with a fourth-order compact finite difference formula, is second order accurate in time and fourth order accurate in space. A linearly stabilized splitting scheme is used to remove the restriction of time step. We prove the unconditional stability of our proposed method in analysis. A fast and efficient linear multigrid solver is employed to solve the resulting discrete system. We perform various numerical experiments to confirm the high-order accuracy, unconditional stability and efficiency of our proposed method. In particular, we show two applications of our proposed method: triply-periodic minimal surface and volume inpainting.

An indirect ALE discretization of single fluid plasma without a fast magnetosonic time step restriction

In this paper we present an adjustment to traditional ALE discretizations of resistive MHD where we do not neglect the time derivative of the electric displacement field. This system is referred to variously as a perfect electromagnetic fluid or a single fluid plasma although we refer to the system as Full Maxwell Hydrodynamics (FMHD) in order to evoke its similarities to resistive Magnetohydrodynamics (MHD). Unlike the MHD system the characteristics of this system do not become arbitrarily large in the limit of low densities. In order to take advantage of these improved characteristics of the system we must tightly couple the electromagnetics into the Lagrangian motion and do away with more traditional operator splitting. We provide a number of verification tests to demonstrate both accuracy of the method and an asymptotic preserving (AP) property. In addition we present a prototype calculation of a Z-pinch and find very good agreement between our algorithm and resistive MHD. Further, FMHD leads to a large performance gain (approximately 4.6x speed up) compared to resistive MHD. We unfortunately find our proposed algorithm does not conserve charge leaving us with an open problem.

Diffusion–dispersion numerical discretization for solute transport in 2D transient shallow flows

The 2D solute transport equation can be incorporated into the 2D shallow water equations in order to solve both flow and solute interactions in a coupled system of equations. In order to solve this system, an explicit finite volume scheme based on Roe’s linearization is proposed. Moreover, it is feasible to decouple the solute transport equation from the hydrodynamic system in a conservative way. In this case, the advection part is solved in essence defining a numerical flux, allowing the use of higher order numerical schemes. However, the discretization of the diffusion–dispersion terms have to be carefully analysed. In particular, time-step restrictions linked to the nature of the solute equation itself as well as the numerical diffusion associated to the numerical scheme used are question of interest in this work. These improvements are tested in an analytical case as well as in a laboratory test case with a passive solute (fluorescein) released from a reservoir. Experimental measurements are compared against the numerical results obtained with the proposed model and a sensitivity analysis is carried out, confirming an agreement with the longitudinal coefficients and an underestimation of the transversal ones, respectively.

Numerical prediction of integrated wave loads on crest walls on top of rubble mound structures

Wave loads on crest walls on top of rubble mound structures determine the size of these crest walls. For the design of crest walls some design guidelines exist, but their validity is limited to particular designs of the cross section (berm, no berm, toe, armour layout, protruding crest element, etc). The present work addresses the validation of OpenFoam/waves2Foam for the prediction of integrated forces on crest wall elements against a new set of experimental data in order to obtain a numerical model that can be applied for a wider field of application than the existing empirical guidelines. One key concern for the accurate modelling of wave loads is the spurious entrapment of air between the water surface and structural elements. The solution developed for this problem is a boundary condition that allows for air ventilation, while enforcing a predefined head loss characteristic. Compared to the existing technique of introducing small meshed tubes through the structure, the new method does not lead to excessive time-step limitations and is therefore more efficient (a practical case was accelerated by a factor 20). The new boundary condition is validated against experimental data of forces on bridge decks with girders.Subsequently, the numerical model is validated against experimental data for loads on crest wall elements from new experiments conducted in a wave flume. The comparison between numerical and experimental data is made both in the time domain and as probability of exceedance. Special emphasis is given to the openness of the faces of the crest wall to mimic the effect of mixing of water and air during the wave impact. Finally, the validated model is applied to evaluate the forces on crest walls as a function of the elevation of the crest wall with respect to the still-water level. This effect is of interest, since the level of the crest wall element is only tested to a limited extent in laboratory experiments and the bottom face was mainly wetted or submerged during these tests (existing empirical formulations). The numerical results are compared to an empirical design formulation [Pedersen, 1996] and conclusions on the general applicability of this particular empirical design formulation are presented. The effect of the shape of the wave spectrum on the resulting forces is investigated in a preliminary fashion.

Unconditional stability and error estimates of the modified characteristics FEMs for the time-dependent incompressible MHD equations

This paper focuses on the unconditional stability and convergence of characteristics type methods for the time-dependent incompressible MHD equations. For this purpose, we introduce a new characteristics time-discrete system. The optimal error estimates in L2 and H1 norms for the typical modified characteristics finite element method unconditionally can be deduced, while the whole previous works require certain time-step restrictions. Some numerical experiments document performance of the characteristics type methods for the time-dependent incompressible MHD equations.

Speed-up of SOLPS-ITER code for tokamak edge modeling

Account of drifts and currents dramatically decreases the accessible time step for the integration of time dependent equations of SOLPS-ITER code for edge modeling. Running the code with sophisticated EIRENE Monte-Carlo model for neutrals and large number of fluid equations for multiple ion species makes the computation time unacceptably long. In the paper the main mechanisms leading to the time step limitations caused by drifts are discussed. Several methods of the suppression of these mechanisms are suggested and the results of numerical scheme tests with the applied corrections are presented. Application of these schemes decreases the time of convergence to steady state solution by more than an order of magnitude.

Treatment of solid objects in the Pencil Code using an immersed boundary method and overset grids

ABSTRACTTwo methods for solid body representation in flow simulations available in the Pencil Code are the immersed boundary method and overset grids. These methods are quite different in terms of computational cost, flexibility and numerical accuracy. We present here an investigation of the use of the different methods with the purpose of assessing their strengths and weaknesses. At present, the overset grid method in the Pencil Code can only be used for representing cylinders in the flow. For this task, it surpasses the immersed boundary method in yielding highly accurate solutions at moderate computational costs. This is partly due to local grid stretching and a body-conformal grid, and partly due to the possibility of working with local time step restrictions on different grids. The immersed boundary method makes up the lack of computational efficiency with flexibility in regard to application to complex geometries, due to a recent extension of the method that allows our implementation of it to represent arbitrarily shaped objects in the flow.

An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs

We extend the entropy stable high order nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations presented by Wintermeyer et al. (2017) [46] with a shock capturing technique and a positivity preservation capability to handle dry areas. The scheme preserves the entropy inequality, is well-balanced and works on unstructured, possibly curved, quadrilateral meshes. For the shock capturing, we introduce an artificial viscosity to the equations and prove that the numerical scheme remains entropy stable. We add a positivity preserving limiter to guarantee non-negative water heights as long as the mean water height is non-negative. We prove that non-negative mean water heights are guaranteed under a certain additional time step restriction for the entropy stable numerical interface flux. We implement the method on GPU architectures using the abstract language OCCA, a unified approach to multi-threading languages. We show that the entropy stable scheme is well suited to GPUs as the necessary extra calculations do not negatively impact the runtime up to reasonably high polynomial degrees (around N=7). We provide numerical examples that challenge the shock capturing and positivity properties of our scheme to verify our theoretical findings.