An official website of the United States government
In this paper, we design and analyze second order positive and free energy satisfying schemes for solving diffusion equations with interaction potentials. The semi-discrete scheme is shown to conserve mass, preserve solution positivity, and satisfy a discrete free energy dissipation law for nonuniform meshes. These properties for the fully-discrete scheme (first order in time) remain preserved without a strict restriction on time steps. For the fully second order (in both time and space) scheme, we use a local scaling limiter to restore solution positivity when necessary. It is proved that such limiter does not destroy the second order accuracy. In addition, these schemes are easy to implement, and efficient in simulations over long time. Both one and two dimensional numerical examples are presented to demonstrate the performance of these schemes.
more »
« less
Discontinuous Galerkin Methods with Time-Operators in Their Numerical Traces for Time-Dependent Electromagnetics
Abstract We present a new class of discontinuous Galerkin methods for the space discretization of the time-dependent Maxwell equations whose main feature is the use of time derivatives and/or time integrals in the stabilization part of their numerical traces.These numerical traces are chosen in such a way that the resulting semidiscrete schemes exactly conserve a discrete version of the energy.We introduce four model ways of achieving this and show that, when using the mid-point rule to march in time, the fully discrete schemes also conserve the discrete energy.Moreover, we propose a new three-step technique to devise fully discrete schemes of arbitrary order of accuracy which conserve the energy in time.The first step consists in transforming the semidiscrete scheme into a Hamiltonian dynamical system.The second step consists in applying a symplectic time-marching method to this dynamical system in order to guarantee that the resulting fully discrete method conserves the discrete energy in time.The third and last step consists in reversing the above-mentioned transformation to rewrite the fully discrete scheme in terms of the original variables.
more »
« less
- Award ID(s):
- 1912646
- PAR ID:
- 10402617
- Date Published:
- Journal Name:
- Computational Methods in Applied Mathematics
- Volume:
- 22
- Issue:
- 4
- ISSN:
- 1609-4840
- Page Range / eLocation ID:
- 775 to 796
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
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 ≤ ρh 2 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 .more » « less
-
Curvilinear, multiblock summation-by-parts finite difference operators with the simultaneous approximation term method provide a stable and accurate framework for solving the wave equation in second order form. That said, the standard method can become arbitrarily stiff when characteristic boundary conditions and nonlinear interface conditions are used. Here we propose a new technique that avoids this stiffness by using characteristic variables to “upwind” the boundary and interface treatment. This is done through the introduction of an additional block boundary displacement variable. Using a unified energy, which expresses both the standard as well as characteristic boundary and interface treatment, we show that the resulting scheme has semidiscrete energy stability for the scalar anisotropic wave equation. The theoretical stability results are confirmed with numerical experiments that also demonstrate the accuracy and robustness of the proposed scheme. The numerical results also show that the characteristic scheme has a time step restriction based on standard wave propagation considerations and not the boundary closure.more » « less
-
In this paper, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we form a new system which incorporates the energy evolution process but is still equivalent to the original equations. Such nonlinear system is then discretized in time based on the backward differentiation formulas, resulting in a dynamically regularized Lagrange multiplier (DRLM) method. First- and second-order DRLM schemes are derived and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting accuracy and stability of the numerical solutions. Fully discrete energy stability is also proved with the Marker-and-Cell (MAC) discretization in space. Various numerical experiments in two and three dimensions verify the convergence and energy dissipation as well as demonstrate the accuracy and robustness of the proposed DRLM schemes.more » « less
-
Abstract In this work we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker–Planck equation and the Keller–Segel equations. The two proposed schemes are first-order accurate in time, explicitly solvable, and second-order and fourth-order accurate in space, which are obtained via finite difference implementation of the classical continuous finite element method. The fully discrete schemes are proved to be positivity preserving and energy dissipative: the second-order scheme can achieve so unconditionally while the fourth-order scheme only requires a mild time step and mesh size constraint. In particular, the fourth-order scheme is the first high order spatial discretization that can achieve both positivity and energy decay properties, which is suitable for long time simulation and to obtain accurate steady state solutions.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1515/cmam-2021-0215
