The aim of this paper is to develop high-order methods for solving time-fractional partial differential equations. The proposed high-order method is based on high-order finite element method for ...space and finite difference method for time. Optimal convergence rate
O
(
(
Δ
t
)
2
−
α
+
N
−
r
)
is proved for the
(
r
−
1
)
th-order finite element method
(
r
≥
2
)
.
We consider initial value/boundary value problems for fractional diffusion-wave equation:
∂
t
α
u
(
x
,
t
)
=
L
u
(
x
,
t
)
, where
0
<
α
⩽
2
, where
L is a symmetric uniformly elliptic operator with
...t-independent smooth coefficients. First we establish the unique existence of the weak solution and the asymptotic behavior as the time
t goes to ∞ and the proofs are based on the eigenfunction expansions. Second for
α
∈
(
0
,
1
)
, we apply the eigenfunction expansions and prove (i) stability in the backward problem in time, (ii) the uniqueness in determining an initial value and (iii) the uniqueness of solution by the decay rate as
t
→
∞
, (iv) stability in an inverse source problem of determining
t-dependent factor in the source by observation at one point over
(
0
,
T
)
.
High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy ...the differential operator, initial condition, and boundary conditions. Our algorithm is meshfree, which is key since meshes become infeasible in higher dimensions. Instead of forming a mesh, the neural network is trained on batches of randomly sampled time and space points. The algorithm is tested on a class of high-dimensional free boundary PDEs, which we are able to accurately solve in up to 200 dimensions. The algorithm is also tested on a high-dimensional Hamilton–Jacobi–Bellman PDE and Burgers' equation. The deep learning algorithm approximates the general solution to the Burgers' equation for a continuum of different boundary conditions and physical conditions (which can be viewed as a high-dimensional space). We call the algorithm a “Deep Galerkin Method (DGM)” since it is similar in spirit to Galerkin methods, with the solution approximated by a neural network instead of a linear combination of basis functions. In addition, we prove a theorem regarding the approximation power of neural networks for a class of quasilinear parabolic PDEs.
•We develop a deep learning algorithm for solving high-dimensional PDEs.•The algorithm is meshfree, which is key since meshes become infeasible in higher dimensions.•We accurately solve a class of high-dimensional free boundary PDEs in up to 200 dimensions.•We prove a theorem regarding the approximation power of neural networks for quasilinear PDEs.
High order accurate weighted essentially nonoscillatory (WENO) schemes are relatively new but have gained rapid popularity in numerical solutions of hyperbolic partial differential equations (PDEs) ...and other convection dominated problems. The main advantage of such schemes is their capability to achieve arbitrarily high order formal accuracy in smooth regions while maintaining stable, nonoscillatory, and sharp discontinuity transitions. The schemes are thus especially suitable for problems containing both strong discontinuities and complex smooth solution features. WENO schemes are robust and do not require the user to tune parameters. At the heart of the WENO schemes is actually an approximation procedure not directly related to PDEs, hence the WENO procedure can also be used in many non-PDE applications. In this paper we review the history and basic formulation of WENO schemes, outline the main ideas in using WENO schemes to solve various hyperbolic PDEs and other convection dominated problems, and present a collection of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology, and some non-PDE applications. Finally, we mention a few topics concerning WENO schemes that are currently under investigation.
In this article, we present a new general formula of a multidimensional general integral transform; this new formula covers mostly all types of multi‐integral transforms of the Laplace class. ...Moreover, single, double, and triple integral transforms of the Laplace kind can be derived from the proposed formula. So, the obtained results in this study can be utilized to reduce computations when using these transforms of same category in other studies. Fundamental properties of the new transform, including linearity, uniqueness, and the inverse, are introduced. In addition, we establish the multi‐convolution theorem and some important results related to partial derivatives. To show the applicability of new multidimensional transform, we employ the new proposed transform to investigate the solutions of some partial differential equations and partial integro‐differential equations.
This paper is concerned with the existence of mild solutions for a class of impulsive fractional partial semilinear differential equations. Some errors in Mophou (2010)
2 are corrected, and some ...previous results are generalized.
We present an unconditionally energy stable finite difference scheme for the Modified Phase Field Crystal equation, a generalized damped wave equation for which the usual Phase Field Crystal equation ...is a special degenerate case. The method is based on a convex splitting of a discrete pseudoenergy and is semi-implicit. The equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable, regardless of time step-size. We present a local-in-time error estimate that ensures the pointwise convergence of the scheme.
Simulation of multiphase flow in porous media is crucial for the effective management of subsurface energy and environment-related activities. The numerical simulators used for modeling such ...processes rely on spatial and temporal discretization of the governing mass and energy balance partial-differential equations (PDEs) into algebraic systems via finite-difference/volume/element methods. These simulators usually require dedicated software development and maintenance, and suffer low efficiency from a runtime and memory standpoint for problems with multi-scale heterogeneity, coupled-physics processes or fluids with complex phase behavior. Therefore, developing cost-effective, data-driven models can become a practical choice, and in this work, we choose deep learning approaches as they can handle high dimensional data and accurately predict state variables with strong nonlinearity. In this paper, we describe a gradient-based deep neural network (GDNN) constrained by the physics related to multiphase flow in porous media. We tackle the nonlinearity of flow in porous media induced by rock heterogeneity, fluid properties, and fluid-rock interactions by decomposing the nonlinear PDEs into a dictionary of elementary differential operators. We use a combination of operators to handle rock spatial heterogeneity and fluid flow by advection. Since the augmented differential operators are inherently related to the physics of fluid flow, we treat them as first principles prior knowledge to regularize the GDNN training. We use the example of pressure management at geologic CO2 storage sites, where CO2 is injected in saline aquifers and brine is produced, and apply GDNN to construct a predictive model that is trained with physics-based simulation data and emulates the physics process. We demonstrate that GDNN can effectively predict the nonlinear patterns of subsurface responses, including the temporal and spatial evolution of the pressure and CO2 saturation plumes. We also successfully extend the GDNN to convolutional neural network (CNN), namely gradient-based CNN (GCNN), and validate its capability to improve the prediction accuracy. GDNN has great potential to tackle challenging problems that are governed by highly nonlinear physics and enable the development of data-driven models with higher fidelity.
•A workflow based on Gradient-based deep neural network (GDNN) emulates multiphase flow in porous media with high fidelity.•Differential operators derived from the governing PDEs are used as first principle prior knowledge to regularize the training of GDNN.•GDNN is further successfully extended to image-based approaches such as convolutional neural networks (CNN).
This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend ...on a finite number of random variables. The method consists of a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the "probability error" with respect to the number of Gauss points in each direction of the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.
•Deep auto-regressive dense encoder-decoder surrogate for predicting transient PDEs.•Physics-constrained learning enables the model to learn dynamics without training data.•A Bayesian framework is ...proposed for interpretable uncertainty quantification of the models' predictions at each time-step.•The auto-regressive model is tested on several non-linear PDE systems with features including turbulence and shock discontinuities.
In recent years, deep learning has proven to be a viable methodology for surrogate modeling and uncertainty quantification for a vast number of physical systems. However, in their traditional form, such models can require a large amount of training data. This is of particular importance for various engineering and scientific applications where data may be extremely expensive to obtain. To overcome this shortcoming, physics-constrained deep learning provides a promising methodology as it only utilizes the governing equations. In this work, we propose a novel auto-regressive dense encoder-decoder convolutional neural network to solve and model non-linear dynamical systems without training data at a computational cost that is potentially magnitudes lower than standard numerical solvers. This model includes a Bayesian framework that allows for uncertainty quantification of the predicted quantities of interest at each time-step. We rigorously test this model on several non-linear transient partial differential equation systems including the turbulence of the Kuramoto-Sivashinsky equation, multi-shock formation and interaction with 1D Burgers' equation and 2D wave dynamics with coupled Burgers' equations. For each system, the predictive results and uncertainty are presented and discussed together with comparisons to the results obtained from traditional numerical analysis methods.
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