High-order Local Discontinuous Galerkin Research Articles

A hybrid finite difference level set–implicit mesh discontinuous Galerkin method for multi-layer coating flows

A mathematical model and numerical framework are presented for computing multi-physics multi-layer coating flow dynamics, with applications to the leveling of multi-layer paint films. The algorithm combines finite difference level set methods and high-order accurate sharp-interface implicit mesh discontinuous Galerkin methods to capture a complex set of multi-physics, incorporating Marangoni-driven multi-phase interfacial flow and the transport, mixing, and evaporation of multiple dissolved species. In particular, we develop several numerical methods for this multi-physics problem, including: high-order local discontinuous Galerkin methods for Poisson problems with Robin boundary conditions on implicitly-defined domains, to capture solvent evaporation; finite difference surface gradient methods, to robustly and accurately incorporate Marangoni stresses; and a coupled multi-physics time stepping approach, to incorporate all the different solvers at play including quasi-Newtonian fluid flow. The framework is applicable to an arbitrary number of layers and dissolved species; here, we apply it in a variety of settings, including multi-solvent evaporative paint dynamics, the flow and leveling of multi-layer automobile paint coatings in both 2D and 3D, and an examination of interfacial turbulence within a multi-layer matter cascade. Our results reproduce several phenomena observed in experiment, such as the formation of Marangoni plumes and Bénard cells. We also use the model to study the impact of long-wave deformational surface modes on immersed interfaces as well as the emergence of the final multi-layer film profile.

Stability and Error Estimates of High Order BDF-LDG Discretizations for the Allen–Cahn Equation

We construct high order local discontinuous Galerkin (LDG) discretizations coupled with third and fourth order backward differentiation formulas (BDF) for the Allen–Cahn equation. The numerical discretizations capture the advantages of linearity and high order accuracy in both space and time. We analyze the stability and error estimates of the time third-order and fourth-order BDF-LDG discretizations for numerically solving Allen–Cahn equation respectively. Theoretical analysis shows the stability and the optimal error results of theses numerical discretizations, in the sense that the time step τ requires only a positive upper bound and is independent of the mesh size h. A series of numerical examples show the correctness of the theoretical analysis. Comparison with the first-order numerical discretization illustrates that the high order BDF-LDG discretizations show good performance in solving stiff problems.

High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes

High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes

Optimal Energy Conserving Local Discontinuous Galerkin Methods for Elastodynamics: Semi and Fully Discrete Error Analysis

We present an arbitrary high-order local discontinuous Galerkin (LDG) method with alternating fluxes for solving linear elastodynamics problems in isotropic media. Both the semi-discrete analysis and fully discrete analysis for a leap-frog LDG method are given to show that the proposed method simultaneously enjoys the energy conserving property and optimal convergence rates in both the displacement and stress, when the tensor product polynomials of the degree k are used on Cartesian meshes. Numerical experiments demonstrate that the proposed method has several advantages including the exact energy conservation, slow-growing errors in long time simulation, and subtle dependence on the first Lame parameter $$\lambda $$ .

High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Diffusion Equation

In this article, some high-order local discontinuous Galerkin (LDG) schemes based on some second-order $$ \theta $$ approximation formulas in time are presented to solve a two-dimensional nonlinear fractional diffusion equation. The unconditional stability of the LDG scheme is proved, and an a priori error estimate with $$O(h^{k+1}+\varDelta t^2)$$ is derived, where $$k\geqslant 0$$ denotes the index of the basis function. Extensive numerical results with $$Q^k(k=0,1,2,3)$$ elements are provided to confirm our theoretical results, which also show that the second-order convergence rate in time is not impacted by the changed parameter $$\theta$$ .

Energy conserving local discontinuous Galerkin methods for the improved Boussinesq equation

The Boussinesq-type equations describe the propagation of weakly non-linear long waves in shallow waters and are widely applied to model water waves in shallow seas and harbors. In this paper, we propose a high-order local discontinuous Galerkin method to solve the improved Boussinesq equation, coupled with both explicit leap-frog and implicit midpoint energy-conserving time discretization. The proposed full-discrete method can be shown to conserve the discrete versions of both mass and energy of the continuous solution. The error estimate with optimal order of convergence is provided for the semi-discrete method. Our numerical experiments confirm optimal rates of convergence as well as the mass and energy conserving property, and show that the errors of the numerical solutions do not grow significantly in time due to the energy conserving property. A series of numerical experiments are provided to show that the proposed method has the capability to simulate the interaction between two solitary waves, single wave break-up and blow-up behavior well.

High-order local discontinuous Galerkin method for simulating wormhole propagation

In this paper, we apply local discontinuous Galerkin methods to the compressible wormhole propagation. Optimal error estimates for the pressure, velocity, porosity and concentration in different norms are established on non-uniform grids. Numerical experiments are presented to verify the theoretical analysis and show the good performance of the LDG scheme for compressible wormhole propagation.

Nonstandard Local Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions

This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs. The proposed LDG methods are natural extensions of a narrow-stencil finite difference framework recently proposed by the authors for approximating viscosity solutions. The idea of the methodology is to use multiple approximations of first and second order derivatives as a way to resolve the potential low regularity of the underlying viscosity solution. Consistency and generalized monotonicity properties are proposed that ensure the numerical operator approximates the differential operator. The resulting algebraic system has several linear equations coupled with only one nonlinear equation that is monotone in many of its arguments. The structure can be explored to design nonlinear solvers. This paper also presents and analyzes numerical results for several numerical test problems in two dimensions which are used to gauge the accuracy and efficiency of the proposed LDG methods.

High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation

In this paper, a high-order local discontinuous Galerkin (LDG) method combined with weighted and shifted Grünwald difference (WSGD) approximation is developed and discussed for a Caputo time-fractional subdiffusion equation. The time fractional derivative of order α, 0<α<1, is approximated by a third-order method based on the idea of WSGD operator, while the spatial operator is approximated by the LDG method. Some useful lemmas are first introduced and proved, then the analysis of stability and optimal error estimate O(Δt3+hk+1) are obtained for the LDG method. Extensive numerical results using Pk,k=0,1,2,3, elements are presented to validate our theoretical analysis.

High Order Local Discontinuous Galerkin Methods for the Allen-Cahn Equation: Analysis and Simulation

In this paper, we present a local discontinuous Galerkin (LDG) method for the AllenCahn equation. We prove the energy stability, analyze the optimal convergence rate of k + 1 in L 2 norm and present the (2k + 1)-th order negative-norm estimate of the semidiscrete LDG method for the Allen-Cahn equation with smooth solution. To relax the severe time step restriction of explicit time marching methods, we construct a first order semi-implicit scheme based on the convex splitting principle of the discrete Allen-Cahn energy and prove the corresponding unconditional energy stability. To achieve high order temporal accuracy, we employ the semi-implicit spectral deferred correction (SDC) method. Combining with the unconditionally stable convex splitting scheme, the SDC method can be high order accurate and stable in our numerical tests. To enhance the efficiency of the proposed methods, the multigrid solver is adapted to solve the resulting nonlinear algebraic systems. Numerical studies are presented to confirm that we can achieve optimal accuracy of O(h k+1 ) in L 2 norm and improve the LDG solution from O(h k+1 ) to O(h 2k+1 ) with the

Positivity preserving high-order local discontinuous Galerkin method for parabolic equations with blow-up solutions

In this paper, we apply positivity-preserving local discontinuous Galerkin (LDG) methods to solve parabolic equations with blow-up solutions. This model is commonly used in combustion problems. However, previous numerical methods are mainly based on a second order finite difference method. This is because the positivity-preserving property can hardly be satisfied for high-order ones, leading to incorrect blow-up time and blow-up sets. Recently, we have applied discontinuous Galerkin methods to linear hyperbolic equations involving δ-singularities and obtained good approximations. For nonlinear problems, some special limiters are constructed to capture the singularities precisely. We will continue this approach and study parabolic equations with blow-up solutions. We will construct special limiters to keep the positivity of the numerical approximations. Due to the Dirichlet boundary conditions, we have to modify the numerical fluxes and the limiters used in the schemes. Numerical experiments demonstrate that our schemes can capture the blow-up sets, and high-order approximations yield better numerical blow-up time.

Local Discontinuous Galerkin Methods for One-Dimensional Second Order Fully Nonlinear Elliptic and Parabolic Equations

This paper is concerned with developing accurate and efficient nonstandard discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in the case of one spatial dimension. The primary goal of the paper to develop a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs which are merely continuous functions by definition. In order to capture discontinuities of the first order derivative $$u_x$$ u x of the solution $$u$$ u , two independent functions $$q^-$$ q - and $$q^+$$ q + are introduced to approximate one-sided derivatives of $$u$$ u . Similarly, to capture the discontinuities of the second order derivative $$u_{xx}$$ u x x , four independent functions $$p^{- -}, p^{- +}, p^{+ -}$$ p - - , p - + , p + - , and $$p^{+ +}$$ p + + are used to approximate one-sided derivatives of $$q^-$$ q - and $$q^+$$ q + . The proposed LDG framework, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a given fully nonlinear problem into a mostly linear system of equations where the given nonlinear differential operator must be replaced by a numerical operator which allows multiple value inputs of the first and second order derivatives $$u_x$$ u x and $$u_{xx}$$ u x x . An easy to verify set of criteria for constructing good numerical operators is also proposed. It consists of consistency and generalized monotonicity. To ensure such a generalized monotonicity property, the crux of the construction is to introduce the numerical moment in the numerical operator, which plays a critical role in the proposed LDG framework. The generalized monotonicity gives the LDG methods the ability to select the viscosity solution among all possible solutions. The proposed framework extends a companion finite difference framework developed by Feng and Lewis (J Comp Appl Math 254:81---98, 2013) and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes. Numerical experiments are also presented to demonstrate the accuracy, efficiency and utility of the proposed LDG methods.