We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8, 9, 19... more We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8, 9, 19, 21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair (β_0,β_1) in the numerical flux, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. A sequence of numerical examples are carried out to demonstrate the accuracy and capability of the maximum-principle-satisfying limiter.
In this paper, we combine a high-order Discontinous Galerkin (DG) method and level set method sol... more In this paper, we combine a high-order Discontinous Galerkin (DG) method and level set method solving the interface problem in a complex incompressible flow. The scheme is L2 stable and conservative. It improves the mass conservative property of the level set method. Numerical examples demonstrate the high order accuracy of the method and the high resolution especially when the interface undergoes large topological changes. Local level set technique is applied to improve the efficiency of the method.
In this paper we consider utilizing a residual neural network (ResNet) to solve ordinary differen... more In this paper we consider utilizing a residual neural network (ResNet) to solve ordinary differential equations. Stochastic gradient descent method is applied to obtain the optimal parameter set of weights and biases of the network. We apply forward Euler, Runge-Kutta2 and Runge-Kutta4 finite difference methods to generate three sets of targets training the ResNet and carry out the target study. The well trained ResNet behaves just as its counterpart of the corresponding one-step finite difference method. In particular, we carry out (1) the architecture study in terms of number of hidden layers and neurons per layer to find the optimal ResNet structure; (2) the target study to verify the ResNet solver behaves as accurate as its finite difference method counterpart; (3) solution trajectory simulation. Even the ResNet solver looks like and is implemented in a way similar to forward Euler scheme, its accuracy can be as high as any one step method. A sequence of numerical examples are p...
Motivated by finite volume scheme, a cell-average based neural network method is proposed. The me... more Motivated by finite volume scheme, a cell-average based neural network method is proposed. The method is based on the integral or weak formulation of partial differential equations. A simple feed forward network is forced to learn the solution average evolution between two neighboring time steps. Offline supervised training is carried out to obtain the optimal network parameter set, which uniquely identifies one finite volume like neural network method. Once well trained, the network method is implemented as a finite volume scheme, thus is mesh dependent. Different to traditional numerical methods, our method can be relieved from the explicit scheme CFL restriction and can adapt to any time step size for solution evolution. For Heat equation, first order of convergence is observed and the errors are related to the spatial mesh size but are observed independent of the mesh size in time. The cell-average based neural network method can sharply evolve contact discontinuity with almost ...
We propose a new formula for the nonlinear viscous numerical flux and extend the direct discontin... more We propose a new formula for the nonlinear viscous numerical flux and extend the direct discontinuous Galerkin method with interface correction (DDGIC) of Liu and Yan [1] to compressible Navier-Stokes equations. The new DDGIC framework is based on the observation that the nonlinear diffusion can be represented as a sum of multiple individual diffusion processes corresponding to each conserved variable. A set of direction vectors corresponding to each individual diffusion process is defined and approximated by the average value of the numerical solution at the cell interfaces. The new framework only requires the computation of conserved variables’ gradient, which is linear and approximated by the original direct DG numerical flux formula. The proposed method greatly simplifies the implementation, and thus, can be easily extended to general equations and turbulence models. Numerical experiments with P1, P2, P3 and P4 polynomial approximations are performed to verify the optimal (k +1)...
In this paper we propose a new local discontinuous Galerkin method to directly solve Hamilton-Jac... more In this paper we propose a new local discontinuous Galerkin method to directly solve Hamilton-Jacobi equations. The scheme is a natural extension of the monotone scheme. For the linear case, the method is equivalent to the discontinuous Galerkin method for conservation laws. Thus, stability and error analysis are obtained under the framework of conservation laws. For both convex and nonconvex Hamiltonian, optimal (k+1)-th order of accuracy for smooth solutions are obtained with piecewise k-th order polynomial approximations. The scheme is numerically tested on a variety of one and two dimensional problems. The method works well to capture sharp corners (discontinuous derivatives) and converges to the viscosity solution. AMS subject classification: 35Q53
In this paper, we apply the Fourier analysis technique to investigate superconvergence properties... more In this paper, we apply the Fourier analysis technique to investigate superconvergence properties of the direct disontinuous Galerkin (DDG) method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009), the DDG method with the interface correction (DDGIC) (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010), the symmetric DDG method (Vidden and Yan in Comput Math 31(6):638–662, 2013), and the nonsymmetric DDG method (Yan in J Sci Comput 54(2):663–683, 2013). We also include the study of the interior penalty DG (IPDG) method, due to its close relation to DDG methods. Error estimates are carried out for both $$P^2$$ and $$P^3$$ polynomial approximations. By investigating the quantitative errors at the Lobatto points, we show that the DDGIC and symmetric DDG methods are superior, in the sense of obtaining $$(k+2)$$ th superconvergence orders for both $$P^2$$ and $$P^3$$ approximations. Superconvergence order of $$(k+2)$$ is also observed for the IPDG method with $$P^3$$ polynomial ap...
We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8, 9, 19... more We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8, 9, 19, 21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair (β_0,β_1) in the numerical flux, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. A sequence of numerical examples are carried out to demonstrate the accuracy and capability of the maximum-principle-satisfying limiter.
In this paper, we combine a high-order Discontinous Galerkin (DG) method and level set method sol... more In this paper, we combine a high-order Discontinous Galerkin (DG) method and level set method solving the interface problem in a complex incompressible flow. The scheme is L2 stable and conservative. It improves the mass conservative property of the level set method. Numerical examples demonstrate the high order accuracy of the method and the high resolution especially when the interface undergoes large topological changes. Local level set technique is applied to improve the efficiency of the method.
In this paper we consider utilizing a residual neural network (ResNet) to solve ordinary differen... more In this paper we consider utilizing a residual neural network (ResNet) to solve ordinary differential equations. Stochastic gradient descent method is applied to obtain the optimal parameter set of weights and biases of the network. We apply forward Euler, Runge-Kutta2 and Runge-Kutta4 finite difference methods to generate three sets of targets training the ResNet and carry out the target study. The well trained ResNet behaves just as its counterpart of the corresponding one-step finite difference method. In particular, we carry out (1) the architecture study in terms of number of hidden layers and neurons per layer to find the optimal ResNet structure; (2) the target study to verify the ResNet solver behaves as accurate as its finite difference method counterpart; (3) solution trajectory simulation. Even the ResNet solver looks like and is implemented in a way similar to forward Euler scheme, its accuracy can be as high as any one step method. A sequence of numerical examples are p...
Motivated by finite volume scheme, a cell-average based neural network method is proposed. The me... more Motivated by finite volume scheme, a cell-average based neural network method is proposed. The method is based on the integral or weak formulation of partial differential equations. A simple feed forward network is forced to learn the solution average evolution between two neighboring time steps. Offline supervised training is carried out to obtain the optimal network parameter set, which uniquely identifies one finite volume like neural network method. Once well trained, the network method is implemented as a finite volume scheme, thus is mesh dependent. Different to traditional numerical methods, our method can be relieved from the explicit scheme CFL restriction and can adapt to any time step size for solution evolution. For Heat equation, first order of convergence is observed and the errors are related to the spatial mesh size but are observed independent of the mesh size in time. The cell-average based neural network method can sharply evolve contact discontinuity with almost ...
We propose a new formula for the nonlinear viscous numerical flux and extend the direct discontin... more We propose a new formula for the nonlinear viscous numerical flux and extend the direct discontinuous Galerkin method with interface correction (DDGIC) of Liu and Yan [1] to compressible Navier-Stokes equations. The new DDGIC framework is based on the observation that the nonlinear diffusion can be represented as a sum of multiple individual diffusion processes corresponding to each conserved variable. A set of direction vectors corresponding to each individual diffusion process is defined and approximated by the average value of the numerical solution at the cell interfaces. The new framework only requires the computation of conserved variables’ gradient, which is linear and approximated by the original direct DG numerical flux formula. The proposed method greatly simplifies the implementation, and thus, can be easily extended to general equations and turbulence models. Numerical experiments with P1, P2, P3 and P4 polynomial approximations are performed to verify the optimal (k +1)...
In this paper we propose a new local discontinuous Galerkin method to directly solve Hamilton-Jac... more In this paper we propose a new local discontinuous Galerkin method to directly solve Hamilton-Jacobi equations. The scheme is a natural extension of the monotone scheme. For the linear case, the method is equivalent to the discontinuous Galerkin method for conservation laws. Thus, stability and error analysis are obtained under the framework of conservation laws. For both convex and nonconvex Hamiltonian, optimal (k+1)-th order of accuracy for smooth solutions are obtained with piecewise k-th order polynomial approximations. The scheme is numerically tested on a variety of one and two dimensional problems. The method works well to capture sharp corners (discontinuous derivatives) and converges to the viscosity solution. AMS subject classification: 35Q53
In this paper, we apply the Fourier analysis technique to investigate superconvergence properties... more In this paper, we apply the Fourier analysis technique to investigate superconvergence properties of the direct disontinuous Galerkin (DDG) method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009), the DDG method with the interface correction (DDGIC) (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010), the symmetric DDG method (Vidden and Yan in Comput Math 31(6):638–662, 2013), and the nonsymmetric DDG method (Yan in J Sci Comput 54(2):663–683, 2013). We also include the study of the interior penalty DG (IPDG) method, due to its close relation to DDG methods. Error estimates are carried out for both $$P^2$$ and $$P^3$$ polynomial approximations. By investigating the quantitative errors at the Lobatto points, we show that the DDGIC and symmetric DDG methods are superior, in the sense of obtaining $$(k+2)$$ th superconvergence orders for both $$P^2$$ and $$P^3$$ approximations. Superconvergence order of $$(k+2)$$ is also observed for the IPDG method with $$P^3$$ polynomial ap...
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