Backward Euler Approximation Research Articles

On backward Euler approximations for systems of conservation laws

On backward Euler approximations for systems of conservation laws

Controlling a generalized Fokker–Planck equation via inputs with nonlocal action

This paper concerns an optimal control problem (P) related to a generalized Fokker–Planck equation. Basic properties of the solutions to the generalized FP equation are derived via a semigroup approach in the space H−1(Rd). Problem (P) is proven to be deeply related to a stochastic optimal control problem (PS) for a McKean–Vlasov equation. The existence of an optimal control is obtained for the deterministic problem (P). The existence of an optimal control is established for an approximating optimal control problem (Ph) related to a backward Euler approximation of the generalized FP equation (with a constant discretization step h). One proves under additional hypotheses that “(Ph) converges to (P)” in a certain sense. First order necessary conditions for (Ph) are derived as well.

Efficient numerical approaches with accelerated graphics processing unit (GPU) computations for Poisson problems and Cahn-Hilliard equations.

In this computational paper, we focused on the efficient numerical implementation of semi-implicit methods for models in materials science. In particular, we were interested in a class of nonlinear higher-order parabolic partial differential equations. The Cahn-Hilliard (CH) equation was chosen as a benchmark problem for our proposed methods. We first considered the Cahn-Hilliard equation with a convexity-splitting (CS) approach coupled with a backward Euler approximation of the time derivative and tested the performance against the bi-harmonic-modified (BHM) approach in terms of accuracy, order of convergence, and computation time. Higher-order time-stepping techniques that allow for the methods to increase their accuracy and order of convergence were then introduced. The proposed schemes in this paper were found to be very efficient for 2D computations. Computed dynamics in 2D and 3D are presented to demonstrate the energy-decreasing property and overall performance of the methods for longer simulation runs with a variety of initial conditions. In addition, we also present a simple yet powerful way to accelerate the computations by using MATLAB built-in commands to perform GPU implementations of the schemes. We show that it is possible to accelerate computations for the CH equation in 3D by a factor of 80, provided the hardware is capable enough.

An unconditionally stable finite element scheme for anisotropic curve shortening flow

Based on a recent novel formulation of parametric anisotropic curve shortening flow, we analyse a fully discrete numerical method of this geometric evolution equation. The method uses piecewise linear finite elements in space and a backward Euler approximation in time. We establish existence and uniqueness of a discrete solution, as well as an unconditional stability property. Some numerical computations confirm the theoretical results and demonstrate the practicality of our method.

An over-penalized weak Galerkin method for parabolic interface problems with time-dependent coefficients

Based on the idea of classical discontinuous Galerkin and weak Galerkin finite element methods, we introduce an over-penalized term in the new scheme as a part of stabilization for solving parabolic interface problems. From the double-valued functions defined on interior edges of elements, it is natural to generate jumps of the over-penalized term. An over-penalized weak Galerkin (OPWG) finite element method can be applied very well to interface problems with general imperfect interface. Importantly, the diffusion coefficients of the interface problems depend on both temporal and spacial variables, not only limited in space as usual. With the use of (Pk(K),Pk−1(e),[Pk−1(K)]d) elements, semi-discrete and fully discrete schemes with backward Euler approximation in time are presented, and then the semi-discrete one is analyzed to be unconditionally stable. To analyze error estimates of semi-discrete and fully discrete schemes directly by error equation, we can just have optimal convergence order in energy norm. By virtue of the introduction of an elliptic projection operator, optimal error estimates of those schemes in L2 norm can be proved. Numerical examples are given to validate the efficiency and optimal convergence orders of the new schemes.

Numerical scheme for transient seepage analysis under unsaturated conditions

Unsaturated soil behaviors characterize the failure mechanisms of geotechnical infrastructures with transient seepage conditions. Therefore, an accurate estimate of the unsaturated groundwater flow is vital in improving hazard management and assessment. This study attempts to develop a numerical scheme for 2-D transient analysis under unsaturated conditions. First, the unsaturated groundwater flow was described using the mass conservation law. Then, the Finite Difference Method and Backward Euler approximation were applied for space and time discretization, respectively. Furthermore, the simple Picard iteration was applied to linearize the governing equation. The reliability of the presented method was verified with the analytical solution. The evaluation results demonstrated the sufficiency of the proposed method, quantitatively expressed by the maximum error of 0.04% for opened boundary conditions and 0.15% for closed boundary conditions. The significant advantage of the proposed method is the flexibility with various soil-water characteristic curve models and associated hydraulic conductivity functions, which helps to improve the applicability in practice.

A Posteriori Error Estimates for Numerical Solutions to Hyperbolic Conservation Laws

The paper is concerned with a posteriori error bounds for a wide class of numerical schemes, for $n\times n$ hyperbolic conservation laws in one space dimension. These estimates are achieved by a "post-processing algorithm", checking that the numerical solution retains small total variation, and computing its oscillation on suitable subdomains. The results apply, in particular, to solutions obtained by the Godunov or the Lax-Friedrichs scheme, backward Euler approximations, and the method of periodic smoothing. Some numerical implementations are presented.

Positive Solutions of the Fractional SDEs with Non-Lipschitz Diffusion Coefficient

We study a class of fractional stochastic differential equations (FSDEs) with coefficients that may not satisfy the linear growth condition and non-Lipschitz diffusion coefficient. Using the Lamperti transform, we obtain conditions for positivity of solutions of such equations. We show that the trajectories of the fractional CKLS model with β>1 are not necessarily positive. We obtain the almost sure convergence rate of the backward Euler approximation scheme for solutions of the considered SDEs. We also obtain a strongly consistent and asymptotically normal estimator of the Hurst index H>1/2 for positive solutions of FSDEs.

Continuous-time reinforcement learning for robust control under worst-case uncertainty

ABSTRACT Reinforcement learning (RL) is an effective method to design a robust controller for unknown nonlinear systems. Uncertainty in the worst case requires a large state-action space. Hence, it is natural to use continuous-time RL methods rather than the discretisation of the spaces. In this paper, we propose a novel continuous-time RL using neural network approximation. Our method uses worst-case uncertainty to train the continuous-time RL algorithm. The backward Euler approximation is used to approximate the time derivative of the value function. Compared with the actor–critic (AC) algorithm, our method finds the robust control policy in the presence of worst-case uncertainty by taking into account the applied actions. It is shown that the AC algorithm finds the robust controller in less episodes, but its robustness is less than the results presented by our approach. The convergence of the proposed algorithm is analysed using the contraction property and differential equation techniques. The experiments show that our approach is more robust than the model-based LQR method and the well-known AC method.

A family of new globally convergent linearization schemes for solving Richards' equation

The Richards' equation models the water flow through porous media in groundwater aquifers and petroleum reservoir simulations. This is a degenerate parabolic differential equation, in which no analytical solution is known. Most numerical methods use implicit time schemes supporting large time steps, but involving near degenerate nonlinear systems. To solve them, we need robust and efficient linearization schemes. Apart from Newton's method, which fails to converge for large time steps and small mesh sizes, recently the globally convergent first order L-scheme was developed, which approximates the derivative by a global upper bound improving Picard's scheme.In this paper, we extend this method to build a family of robust efficient globally convergent first-order linearization schemes by using a sequence of real functions Ln, n≥0. We solve the space variable by a piecewise linear Galerkin finite element method. The time discretization is based on the Forward-Backward Euler method in the term of hydraulic conductivity K, whereas the remaining terms of the equation dealt with the Backward Euler approximation. We get five schemes improving the convergence of L-schemes and are getting closer to a second-order convergent method. We prove the time iteration convergence in the H1(Ω) norm, and we test the schemes on numerical benchmarks to compare them with the L-scheme and Newton's scheme. Among the new schemes, four of them are globally convergent and one scheme is a hybrid case with Newton's scheme. Our results show that among the new schemes, Modified Generalized Linear Scheme (MGLS) obtains the best convergence rates (up to four times that of the L-scheme) using fewer iteration steps and less overall computing time. Finally, our hybrid scheme is robust since it converges when Newton's scheme does not and uses similar computing time, which makes it a good alternative for locally convergent second-order schemes. More methods can be developed using this framework with other Ln sequences and spatial discretizations.

Vanishing Viscosity and Backward Euler Approximations for Conservation Laws with Discontinuous Flux

Solutions to a class of one-dimensional conservation laws with discontinuous flux are constructed relying on the Crandall--Liggett theory of nonlinear contractive semigroups [H. Brezis and A. Pazy,...

Investigation of Lagrange–Galerkin Method for an Obstacle Parabolic Problem

The convergence and accuracy estimates are proved for Lagrange–Galerkin method, used for approximating the parabolic obstacle problem. The convergence analysis is based on the comparison of the solutions of Lagrange–Galerkin and backward Euler approximation schemes. First order in time step estimate for the difference of the solutions for above schemes in energy norm is proved under sufficiently weak requirements for the smoothness of the initial data. First order in time and space steps accuracy estimate for Lagrange–Galerkin method is derived in the case of discontinuous time derivative of the exact solution.

Positivity preserving finite volume scheme for the Nagumo-type equations on distorted meshes

In this paper we present a nonlinear positivity preserving finite volume scheme for the Nagumo-type equations with anisotropic tensor diffusion coefficient. For the diffusion term, we use the positivity preserving finite volume scheme. For the time direction, we use the backward Euler approximation. We deal with nonlinear reaction term implicitly and decompose nonlinear reaction coefficient into two nonnegative functions. Thus we get a system of nonlinear algebraic equations. The advantages of our scheme are that it can be applied to distorted meshes and has no severe constraint on the time step. The numerical results verify the theoretical result.

Numerical computation of triangular cavity flows by a Lagrange-Galerkin scheme with a locally linearized velocity

We show numerical results of triangular cavity flow problems solved by a Lagrange-Galerkin scheme free from numerical quadrature. The scheme has recently developed by us, where a locally linearized velocity and the backward Euler approximation are used in finding the position of fluid particle at the previous time step. Since the scheme can be implemented exactly as it is, the theoretical stability and convergence results are assured, while the conventional Lagrange-Galerkin schemes may encounter the instability caused by numerical quadrature errors. The scheme is employed to solve cavity flow problems in triangular domains, where we observe the bifurcation of stationary solutions and the patterns of streamlines.

A Posteriori Error Estimates for Lumped Mass Finite Element Method for Linear Parabolic Problems Using Elliptic Reconstruction

ABSTRACTWe study residual-based a posteriori error estimates for both the spatially discrete and the fully discrete lumped mass finite element approximation for parabolic problems in a bounded convex polygonal domain in ℝ2. In particular, the space discretization uses finite element spaces that are assumed to be nested one and the time discretization is based on the backward Euler approximation. The main key features used in the analysis are the reconstruction technique and energy argument combined with the stability of L2 projection in H1(Ω). The a posteriori error estimates we derive are optimal order in both the and -norms.

High‐accuracy multi‐rate implementation of resonant integrator using FPGA

Second-order generalised integrator or resonant integrator (RI) has wide range of applications. Forward and backward Euler's approximation based two integrator realisation of RI is an easily implementable frequency adaptive method. However, it suffers from resonant frequency deviation due to discretisation. The discretisation methods that lead to accurate realisation of RI require online calculation or lookup table of trigonometric functions to accommodate frequency variation. In this study, multi-rate computation-based implementation of two integrator-based RI has been proposed to minimise resonant frequency deviation. In this method, no additional logic elements are consumed to achieve accurate resonant frequency location. This along with down-sampling leads to lesser phase lag of RI. The effect of quantisation on resonant frequency deviation has been analysed for proper choice of calculation time. It is also shown that appropriate choice of down-sampling instants give a range of phase response characteristics around the nominal continuous time RI phase response. The accuracy of resonant frequency emulation has been experimentally verified by implementing a proportional–resonant controller as current controller.

New interpolation error estimates and a posteriori error analysis for linear parabolic interface problems

We derive residual‐based a posteriori error estimates of finite element method for linear parabolic interface problems in a two‐dimensional convex polygonal domain. Both spatially discrete and fully discrete approximations are analyzed. While the space discretization uses finite element spaces that are allowed to change in time, the time discretization is based on the backward Euler approximation. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates and an appropriate adaptation of the elliptic reconstruction technique introduced by (Makridakis and Nochetto, SIAM J Numer Anal 4 (2003), 1585–1594). We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the ‐norm and almost optimal order in the ‐norm. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical results are presented to validate our derived estimators. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 570–598, 2017

The well-posedness of a nonlocal hyperbolic model for type-I superconductors

A vectorial nonlocal linear hyperbolic problem with applications in superconductors of type-I is studied. The nonlocal term is represented by a (space) convolution with a singular kernel, which is arising in Eringen's model. The well-posedness of the problem is discussed under low regularity assumptions and the error estimates for two time-discrete schemes (based on backward Euler approximation) are established.

A semigroup approach to an integro-differential equation modeling slow erosion

The paper is concerned with a scalar conservation law with nonlocal flux, providing a model for granular flow with slow erosion and deposition. While the solution u=u(t,x) can have jumps, the inverse function x=x(t,u) is always Lipschitz continuous; its derivative has bounded variation and satisfies a balance law with measure-valued sources. Using a backward Euler approximation scheme combined with a nonlinear projection operator, we construct a continuous semigroup whose trajectories are the unique entropy weak solutions to this balance law. Going back to the original variables, this yields the global well-posedness of the Cauchy problem for the granular flow model.

A nonlocal parabolic model for type‐I superconductors

A vectorial nonlocal linear parabolic problem on a bounded domain with applications in superconductors of type‐I is studied. The nonlocal term is represented by a (space) convolution with a singular kernel (arising in Eringen's model). The well‐posedness of the problem is discussed under low regularity assumptions, and the error estimates for an implicit and semiimplicit time‐discrete scheme (based on backward Euler approximation) are established. It is shown that the solution of the problem satisfies a simpler nonlocal problem with a positive definite kernel if the normal component of the unknown vector field equals zero on the boundary of the domain. Numerical experiments support the obtained theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1821–1853, 2014