Explicit Time Discretization Research Articles

Three-dimensional control-volume distributed multi-point flux approximation coupled with a lower-dimensional surface fracture model

A novel cell-centred control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulation is presented for discrete fracture–matrix simulations on unstructured grids in three-dimensions (3D). The grid is aligned with fractures and barriers which are then modelled as lower-dimensional surface interfaces located between the matrix cells in the physical domain. The three-dimensional pressure equation is solved in the matrix domain coupled with a two-dimensional (2D) surface pressure equation solved over fracture networks via a novel surface CVD-MPFA formulation. The CVD-MPFA formulation naturally handles fractures with anisotropic permeabilities on unstructured grids. Matrix–fracture fluxes are expressed in terms of matrix and fracture pressures and define the transfer function, which is added to the lower-dimensional flow equation and couples the three-dimensional and surface systems. An additional transmission condition is used between matrix cells adjacent to low permeable fractures to couple the velocity and pressure jump across the fractures. Convergence and accuracy of the lower-dimensional fracture model is assessed for highly anisotropic fractures having a range of apertures and permeability tensors. A transport equation for tracer flow is coupled via the Darcy flux for single and intersecting fractures. The lower-dimensional approximation for intersecting fractures avoids the more restrictive CFL condition corresponding to the equi-dimensional approximation with explicit time discretisation. Lower-dimensional fracture model results are compared with equi-dimensional model results. Fractures and barriers are efficiently modelled by lower-dimensional interfaces which yield comparable results to those of the equi-dimensional model. Pressure continuity is built into the model across highly conductive fractures, leading to reduced local degrees of freedom in the CVD-MPFA approximation. The formulation is applied to geologically complex fracture networks in three-dimensions. The effects of the fracture permeability, aperture and grid resolution are also assessed with respect to convergence and computational cost.

Iterative solvers for the Maxwell–Stefan diffusion equations: Methods and applications in plasma and particle transport

In this paper, we are motivated to discuss a model based on a local thermodynamic equilibrium, weakly ionized plasma-mixture model used for medical and technical applications in etching processes. For studying the model, we consider a simplified model based on the Maxwell–Stefan model, which describes multicomponent diffusive fluxes in the gas mixture. The MS model is more adequate to describe complex mixtures without dominating background species. Based on additional conditions to the fluxes, we obtain an irreducible and quasi-positive diffusion matrix. Such problems result in nonlinear diffusion equations, which are more delicate to solve as simpler standard diffusion equations with Fickian’s approach. Here, we propose an efficient explicit time-discretization method, which is embedded to a fast iterative solver for the nonlinearities. Such a combination of coupling discretization and solver methods allows to simulate the delicate nonlinear differential equations more effectively. We present the efficie...

An efficient, unconditionally energy stable local discontinuous Galerkin scheme for the Cahn–Hilliard–Brinkman system

In this paper, we present an efficient and unconditionally energy stable fully-discrete local discontinuous Galerkin (LDG) method for approximating the Cahn–Hilliard–Brinkman (CHB) system, which is comprised of a Cahn–Hilliard type equation and a generalized Brinkman equation modeling fluid flow. The semi-discrete energy stability of the LDG method is proved firstly. Due to the strict time step restriction (Δt=O(Δx4)) of explicit time discretization methods for stability, we introduce a semi-implicit scheme which consists of the implicit Euler method combined with a convex splitting of the discrete Cahn–Hilliard energy strategy for the temporal discretization. The unconditional energy stability of this fully-discrete convex splitting scheme is also proved. Obviously, the fully-discrete equations at the implicit time level are nonlinear, and to enhance the efficiency of the proposed approach, the nonlinear Full Approximation Scheme (FAS) multigrid method has been employed to solve this system of algebraic equations. We also show the nearly optimal complexity numerically. Numerical experiments based on the overall solution method of combining the proposed LDG method, convex splitting scheme and the nonlinear multigrid solver are given to validate the theoretical results and to show the effectiveness of the proposed approach for the CHB system.

A scalable coupled surface–subsurface flow model

The coupling of physically-based models for surface and subsurface water flows is a recent concern. The study of their interactions is important both for water resource management and environmental studies. However, despite constant innovation, physically-based simulations of water flows are still time consuming. That is especially problematic for large and/or long-term studies, or to test a large range of parametrizations with an adjoint model. As the current trend in computing sciences is to increase the computational power with additional computational units, new model developments are expected to scale efficiently on parallel infrastructures. This paper describes a coupled surface–subsurface flow model that combines implicit and explicit time discretizations for the surface and subsurface dynamics, respectively. Despite that the surface flow has a faster dynamics than the subsurface flow, we are able to use a unique nearly-optimal time step for each submodel, hence improving the resources use. The surface model is discretized with an implicit control volume finite element method while the subsurface model is solved by means of an explicit discontinuous Galerkin finite element method. The surface and subsurface models are coupled by weakly imposing the continuity of water pressure. By imposing a threshold on the influence coefficients of the control volume finite element method, we can prevent the occurrence of unphysical fluxes in anisotropic elements. The proposed coupling is shown to produce results similar to state-of-the-art models for four different test cases while achieving better strong and weak scalings on up to 192 processors.

Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for nonlinear convection–diffusion problems

The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with implicit–explicit (IMEX) time discretization schemes, for solving one-dimensional convection–diffusion equations with a nonlinear convection. Both Runge–Kutta and multi-step IMEX methods are considered. By the aid of the energy method, we show that the IMEX LDG schemes are unconditionally stable for the nonlinear problems, in the sense that the time-step τ is only required to be upper-bounded by a positive constant which depends on the flow velocity and the diffusion coefficient, but is independent of the mesh size h. We also give optimal error estimates for the IMEX LDG schemes, under the same temporal condition, if a monotone numerical flux is adopted for the convection. Numerical experiments are given to verify our main results.

Control-volume distributed multi-point flux approximation coupled with a lower-dimensional fracture model

A cell-centered control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulation is presented for discrete fracture-matrix simulations. The grid is aligned with the fractures and barriers which are then modeled as lower-dimensional interfaces located between the matrix cells in the physical domain. The nD pressure equation is solved in the matrix domain coupled with an (n−1)D pressure equation solved in the fractures. The CVD-MPFA formulation naturally handles fractures with anisotropic permeabilities on unstructured grids. Matrix-fracture fluxes are expressed in terms of matrix and fracture pressures, and must be added to the lower-dimensional flow equation (called the transfer function). An additional transmission condition is used between matrix cells adjacent to low permeable fractures to link the velocity and pressure jump across the fractures. Numerical tests serve to assess the convergence and accuracy of the lower-dimensional fracture model for highly anisotropic fractures having different apertures and permeability tensors. A transport equation for tracer flow is coupled via the Darcy flux for single and intersecting fractures. The lower-dimensional approach for intersecting fractures avoids the more restrictive CFL condition corresponding to the equi-dimensional approximation with explicit time discretization. Lower-dimensional fracture model results are compared with hybrid-grid and equi-dimensional model results. Fractures and barriers are efficiently modeled by lower-dimensional interfaces which yield comparable results to those of the equi-dimensional model. Highly conductive fractures are modeled as lower-dimensional entities without the use of locally refined grids that are required by the equi-dimensional model, while pressure continuity across fractures is built into the model, without depending on the extra degrees of freedom which must be added locally by the hybrid-grid method. The lower-dimensional fracture model also yields improved results when compared to those of the hybrid-grid model for fractures with low-permeability in the normal direction to the fracture. In addition, a transient pressure simulation involving geologically representative complex fracture networks is presented.

A novel 1D/2D model for simulating conjugate heat transfer applied to flow boiling in tubes with external fins

This study presents a novel, simplified model for the time-efficient simulation of transient conjugate heat transfer in round tubes. The flow domain and the tube wall are modeled in 1D and 2D, respectively and empirical correlations are used to model the flow domain in 1D. The model is particularly useful when dealing with complex physics, such as flow boiling, which is the main focus of this study. The tube wall is assumed to have external fins. The flow is vertical upwards. Note that straightforward computational fluid dynamics (CFD) analysis of conjugate heat transfer in a system of tubes, leads to 3D modeling of fluid and solid domains. Because correlation is used and dimensionality reduced, the model is numerically more stable and computationally more time-efficient compared to the CFD approach. The benefit of the proposed approach is that it can be applied to large systems of tubes as encountered in many practical applications. The modeled equations are discretized in space using the finite volume method, with central differencing for the heat conduction equation in the solid domain, and upwind differencing of the convective term of the enthalpy transport equation in the flow domain. An explicit time discretization with forward differencing was applied to the enthalpy transport equation in the fluid domain. The conduction equation in the solid domain was time discretized using the Crank–Nicholson scheme. The model is applied in different boundary conditions and the predicted boiling patterns and temperature fields are discussed.

Operator Bounds and Time Step Conditions for the DG and Central DG Methods

Discontinuous Galerkin (DG) and central DG methods are two related families of finite element methods. They can provide high order spatial discretizations that are often combined with explicit high order time discretizations to solve initial boundary value problems. In this context, it has been observed that the central DG method allows larger time steps, especially for schemes with high accuracy. In this paper, we estimate bounds for the DG and central DG spatial operators for the linear advection equation. Based on these estimates and Kreiss-Wu theory, we obtain time step conditions to ensure the numerical stability of the DG and central DG methods when the methods are combined with locally stable time discretizations. In particular, for a fixed time discretization, the time step allowed for the DG method is proportional to $$h/k^2$$ h / k 2 , while the time step allowed for the central DG method is proportional to $$h/k$$ h / k , where $$h$$ h is the spatial mesh size and $$k>0$$ k > 0 is the polynomial degree of the discrete space of the spatial discretization. In addition, the analysis provides new insight into the role of a parameter in the central DG formulation. We verify our results numerically, and we also discuss extensions of our analysis to some related discretizations.

Unsaturated subsurface flow with surface water and nonlinear in- and outflow conditions

We analytically and numerically analyze groundwater flow in a homogeneous soil described by the Richards equation, coupled to surface water represented by a set of ordinary differential equations (ODEs) on parts of the domain boundary, and with nonlinear outflow conditions of Signorini's type. The coupling of the partial differential equation (PDE) and the ODE's is given by nonlinear Robin boundary conditions. This paper provides two major new contributions regarding these infiltration conditions. First, an existence result for the continuous coupled problem is established with the help of a regularization technique. Second, we analyze and validate a solver-friendly discretization of the coupled problem based on an implicit–explicit time discretization and on finite elements in space. The discretized PDE leads to convex spatial minimization problems which can be solved efficiently by monotone multigrid. Numerical experiments are provided using the DUNE numerics framework.

An efficient fully-discrete local discontinuous Galerkin method for the Cahn–Hilliard–Hele–Shaw system

In this paper, we develop an efficient and energy stable fully-discrete local discontinuous Galerkin (LDG) method for the Cahn–Hilliard–Hele–Shaw (CHHS) system. The semi-discrete energy stability of the LDG method is proved firstly. Due to the strict time step restriction (Δt=O(Δx4)) of explicit time discretization methods for stability, we introduce a semi-implicit time integration scheme which is based on a convex splitting of the discrete Cahn–Hilliard energy. The unconditional energy stability has been proved for this fully-discrete LDG scheme. The fully-discrete equations at the implicit time level are nonlinear. Thus, the nonlinear Full Approximation Scheme (FAS) multigrid method has been applied to solve this system of algebraic equations, which has been shown the nearly optimal complexity numerically. Numerical results are also given to illustrate the accuracy and capability of the LDG method coupled with the multigrid solver.

Generalized Multiscale Finite Element Methods for Wave Propagation in Heterogeneous Media

Numerical modeling of wave propagation in heterogeneous media is important in many applications. Due to their complex nature, direct numerical simulations on the fine grid are prohibitively expensive. It is therefore important to develop efficient and accurate methods that allow the use of coarse grids. In this paper, we present a multiscale finite element method for wave propagation on a coarse grid. The proposed method is based on the generalized multiscale finite element method (GMsFEM) (see [Y. Efendiev, J. Galvis, and T. Hou, J. Comput. Phys., 251 (2012), pp. 116--135]). To construct multiscale basis functions, we start with two snapshot spaces in each coarse-grid block, where one represents the degrees of freedom on the boundary and the other represents the degrees of freedom in the interior. We use local spectral problems to identify important modes in each snapshot space. These local spectral problems are different from each other and their formulations are based on the analysis. To the best of knowledge, this is the first time that multiple snapshot spaces and multiple spectral problems are used and necessary for efficient computations. Using the dominant modes from local spectral problems, multiscale basis functions are constructed to represent the solution space locally within each coarse block. These multiscale basis functions are coupled via the symmetric interior penalty discontinuous Galerkin method which provides a block diagonal mass matrix and, consequently, results in fast computations in an explicit time discretization. Our methods' stability and spectral convergence are rigorously analyzed. Numerical examples are presented to show our methods' performance. We also test oversampling strategies. In particular, we discuss how the modes from different snapshot spaces can affect the proposed methods' accuracy.

Extended generalized Lagrangian multipliers for magnetohydrodynamics using adaptive multiresolution methods

We present a new adaptive multiresoltion method for the numerical simulation of ideal magnetohydrodynamics. The governing equations, i.e. , the compressible Euler equations coupled with the Maxwell equations are discretized using a finite volume scheme on a two-dimensional Cartesian mesh. Adaptivity in space is obtained via Harten’s cell average multiresolution analysis, which allows the reliable introduction of a locally refined mesh while controlling the error. The explicit time discretization uses a compact Runge–Kutta method for local time stepping and an embedded Runge-Kutta scheme for automatic time step control. An extended generalized Lagrangian multiplier approach with the mixed hyperbolic-parabolic correction type is used to control the incompressibility of the magnetic field. Applications to a two-dimensional problem illustrate the properties of the method. Memory savings and numerical divergences of magnetic field are reported and the accuracy of the adaptive computations is assessed by comparing with the available exact solution.

On the numerical solution to a nonlinear wave equation associated with the first painlevé equation: an operator-splitting approach

The main goal of this article is to discuss the numerical solution to a nonlinear wave equation associated with the first of the celebrated Painleve transcendent ordinary differential equations. In order to solve numerically the above equation, whose solutions blow up in finite time, the authors advocate a numerical methodology based on the Strang’s symmetrized operator-splitting scheme. With this approach, one can decouple nonlinearity and differential operators, leading to the alternate solution at every time step of the equation as follows: (i) The first Painleve ordinary differential equation, (ii) a linear wave equation with a constant coefficient. Assuming that the space dimension is two, the authors consider a fully discrete variant of the above scheme, where the space-time discretization of the linear wave equation sub-steps is achieved via a Galerkin/finite element space approximation combined with a second order accurate centered time discretization scheme. To handle the nonlinear sub-steps, a second order accurate centered explicit time discretization scheme with adaptively variable time step is used, in order to follow accurately the fast dynamic of the solution before it blows up. The results of numerical experiments are presented for different coefficients and boundary conditions. They show that the above methodology is robust and describes fairly accurately the evolution of a rather “violent” phenomenon.

A NEW MONTE CARLO METHOD FOR TIME-DEPENDENT NEUTRINO RADIATION TRANSPORT

Monte Carlo approaches to radiation transport have several attractive properties such as simplicity of implementation, high accuracy, and good parallel scaling. Moreover, Monte Carlo methods can handle complicated geometries and are relatively easy to extend to multiple spatial dimensions, which makes them potentially interesting in modeling complex multi-dimensional astrophysical phenomena such as core-collapse supernovae. The aim of this paper is to explore Monte Carlo methods for modeling neutrino transport in core-collapse supernovae. We generalize the Implicit Monte Carlo photon transport scheme of Fleck & Cummings and gray discrete-diffusion scheme of Densmore et al. to energy-, time-, and velocity-dependent neutrino transport. Using our 1D spherically-symmetric implementation, we show that, similar to the photon transport case, the implicit scheme enables significantly larger timesteps compared with explicit time discretization, without sacrificing accuracy, while the discrete-diffusion method leads to significant speed-ups at high optical depth. Our results suggest that a combination of spectral, velocity-dependent, Implicit Monte Carlo and discrete-diffusion Monte Carlo methods represents a robust approach for use in neutrino transport calculations in core-collapse supernovae. Our velocity-dependent scheme can easily be adapted to photon transport.

A one-dimensional shock-capturing model for long wave run-up on sloping beaches

This paper presents a one-dimensional finite-volume model to investigate long wave run-up under non-breaking and breaking conditions. A conservative form of the non-linear shallow water equations is solved using a high-resolution Godunov-type scheme coupled with the HLL Riemann solver. The surface gradient method leads to a well-balanced formulation between the flux and the source terms. The explicit time discretisation is first-order accurate, but a piecewise linear reconstruction of numerical data at cell interfaces helps achieve second-order accuracy in space. The computed surface elevation, flow velocity and run-up show very good agreement with available analytical solutions and experimental observations. The model accurately describes propagation of non-breaking and breaking waves on a sloping beach.

AN ENERGY-CONSERVING DISCONTINUOUS MULTISCALE FINITE ELEMENT METHOD FOR THE WAVE EQUATION IN HETEROGENEOUS MEDIA

Seismic data are routinely used to infer in situ properties of earth materials on many scales, ranging from global studies to investigations of surficial geological formations. While inversion and imaging algorithms utilizing these data have improved steadily, there are remaining challenges that make detailed measurements of the properties of some geologic materials very difficult. For example, the determination of the concentration and orientation of fracture systems is prohibitively expensive to simulate on the fine grid and, thus, some type of coarse-grid simulations are needed. In this paper, we describe a new multiscale finite element algorithm for simulating seismic wave propagation in heterogeneous media. This method solves the wave equation on a coarse grid using multiscale basis functions and a global coupling mechanism to relate information between fine and coarse grids. Using a mixed formulation of the wave equation and staggered discontinuous basis functions, the proposed multiscale methods have the following properties. • The total wave energy is conserved. • Mass matrix is diagonal on a coarse grid and explicit energy-preserving time discretization does not require solving a linear system at each time step. • Multiscale basis functions can accurately capture the subgrid variations of the solution and the time stepping is performed on a coarse grid. We discuss various subgrid capturing mechanisms and present some preliminary numerical results.

Including the effects of temperature-dependent opacities in the implicit Monte Carlo algorithm

The Implicit Monte Carlo technique of Fleck and Cummings [1] is often employed to numerically simulate radiative transfer. This method achieves greater stability than one with a fully explicit time discretization by estimating the t n+1 value of T 4 from the thermal emission term, which is proportional to T 4. In the Fleck and Cummings algorithm, this results in decreasing the absorption by the so-called “Fleck factor”, and adding a corresponding amount of effective scattering. We show how to include the effects of the temperature-dependent opacity to the estimated t n+1 value of the thermal emission term. This results in the addition to the “Fleck factor” of a term that depends on d σ d T . We demonstrate that this modification allows for more accurate solutions with much larger time steps for problems with opacities that have a strong temperature dependence.

Solution of Burgers’ Equation Using a Local-RBF Meshless Method

A local radial basis function (RBF) meshless method is applied for solution of the Burgers’ equation with different initial and boundary conditions of various complexities. Local-RBF collocation is employed for discretization in space, whilst the unsteady term is handled via a simple explicit time discretization. Moreover, in case of non-smooth initial conditions with high Reynolds numbers, a treatment is proposed for inability of local-RBF methods to solve such problems. The scheme is validated over a variety of benchmark problems and very good agreement is found with existing analytical and numerical solutions.

Partitioned time discretization for parallel solution of coupled ODE systems

One decoupling method for multiphysics, multiscale, multidomain applications involves partitioning the problem via explicit time discretizations in the coupling terms. Specialized, problem-specific techniques are needed for the resulting partitioned methods to avoid time step restrictions which make long time calculations costly. This report studies unconditionally stable, uncoupled time stepping methods for a model problem sharing mathematical structure akin to the coupled atmosphere-ocean system. Three decoupled time stepping algorithms are introduced and their stability and consistency are rigorously examined. Numerical experiments are performed that study their stability and convergence properties.

Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors

We consider a family of explicit one-step time discretizations for finite volume and discontinuous Galerkin schemes, which is based on a predictor-corrector formulation. The predictor remains local taking into account the time evolution of the data only within the grid cell. Based on a space–time Taylor expansion, this idea is already inherent in the MUSCL finite volume scheme to get second order accuracy in time and was generalized in the context of higher order ENO finite volume schemes. We interpret the space–time Taylor expansion used in this approach as a local predictor and conclude that other space–time approximate solutions of the local Cauchy problem in the grid cell may be applied. Three possibilities are considered in this paper: (1) the classical space–time Taylor expansion, in which time derivatives are obtained from known space-derivatives by the Cauchy–Kovalewsky procedure; (2) a local continuous extension Runge–Kutta scheme and (3) a local space–time Galerkin predictor with a version suitable for stiff source terms. The advantage of the predictor–corrector formulation is that the time evolution is done in one step which establishes optimal locality during the whole time step. This time discretization scheme can be used within all schemes which are based on a piecewise continuous approximation as finite volume schemes, discontinuous Galerkin schemes or the recently proposed reconstructed discontinuous Galerkin or P N P M schemes. The implementation of these approaches is described, advantages and disadvantages of different predictors are discussed and numerical results are shown.