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Bayesian Physics Informed Neural Networks (B-PINNs) have gained significant attention for inferring physical parameters and learning the forward solutions for problems based on partial differential equations. However, the overparameterized nature of neural networks poses a computational challenge for high-dimensional posterior inference. Existing inference approaches, such as particle-based or variance inference methods, are either computationally expensive for high-dimensional posterior inference or provide unsatisfactory uncertainty estimates. In this paper, we present a new efficient inference algorithm for B-PINNs that uses Ensemble Kalman Inversion (EKI) for high-dimensional inference tasks. By reframing the setup of B-PINNs as a traditional Bayesian inverse problem, we can take advantage of EKI's key features: (1) gradient-free, (2) computational complexity scales linearly with the dimension of the parameter spaces, and (3) rapid convergence with typically O (100) iterations. We demonstrate the applicability and performance of the proposed method through various types of numerical examples. We find that our proposed method can achieve inference results with informative uncertainty estimates comparable to Hamiltonian Monte Carlo (HMC)-based B-PINNs with a much reduced computational cost. These findings suggest that our proposed approach has great potential for uncertainty quantification in physics-informed machine learning for practical applications. • Proposed a gradient-free inference algorithm for Bayesian physics informed neural networks (BPINNs). • Reframed BPINNs as a classic inverse problem to utilize Ensemble Kalman Inversion (EKI). • Demonstrated our method achieves informative uncertainty estimates at lower cost. [ABSTRACT FROM AUTHOR]

Free-surface, mobile-bed, shallow-flow models may present Hybrid hyperbolic systems of partial differential equations characterised by conservative and non-conservative fluxes that can only be expressed in primitive variables. This paper presents the effort we made to derive DOT-type schemes (Osher-type schemes derived by Dumbser and Toro, 2011 [1]) for these kinds of systems formulated for the one-dimensional case. Firstly, for a Hybrid system, we managed to write a quasi-linear form characterised by the presence of a matrix, expressed as a function of the primitive variables, that multiplies the spatial derivative of the conserved variables. Next, we derived the first numerical flux by adapting the approach of Leibinger et al., 2016 [2] to this quasi-linear form. We called this result DOT HCP flux. To achieve a faster algorithm, instead of using an integration path in the space of conserved variables, as in the previous case, we employed a path in the space of primitive variables. We called this second formulation DOT HPP flux. Subsequently, we managed to account for certain physical constraints arising from the generalised Rankine-Hugoniot relations in the expression of one term of the previous flux formulation, thus obtaining the DOT HZR flux. Finally, we showed that these methods can also be applied to Combined systems characterised by conservative and non-conservative fluxes expressed in conserved variables. Several tests show the characteristics and good performances of the proposed methods when applied to Riemann problems of Hybrid and Combined systems deriving from free-surface models. Finally, thanks to the general formulation of the proposed DOT-type fluxes, these can also be applied to Hybrid and Combined hyperbolic systems deriving from different physical problems. [ABSTRACT FROM AUTHOR]

Automatic differentiation (AD) is a critical step in physics-informed machine learning, required for computing the high-order derivatives of network output w.r.t. coordinates of collocation points. In this paper, we present a novel and lightweight algorithm to conduct AD for physics-informed operator learning, which we call the trick of Zero Coordinate Shift (ZCS). Instead of making all sampled coordinates as leaf variables, ZCS introduces only one scalar-valued leaf variable for each spatial or temporal dimension, simplifying the wanted derivatives from "many-roots-many-leaves" to "one-root-many-leaves" whereby reverse-mode AD becomes directly utilisable. It has led to an outstanding performance leap by avoiding the duplication of the computational graph along the dimension of functions (physical parameters). ZCS is easy to implement with current deep learning libraries; our own implementation is achieved by extending the DeepXDE package. We carry out a comprehensive benchmark analysis and several case studies, training physics-informed DeepONets to solve partial differential equations (PDEs) without data. The results show that ZCS has persistently reduced GPU memory consumption and wall time for training by an order of magnitude, and such reduction factor scales with the number of functions. As a low-level optimisation technique, ZCS imposes no restrictions on data, physics (PDE) or network architecture and does not compromise training results from any aspect. • We present a novel algorithm to conduct automatic differentiation w.r.t. coordinates for physics-informed operator learning. • Our algorithm can reduce GPU memory and wall time for training physics-informed DeepONets by an order of magnitude. • Our algorithm neither affects training results nor imposes any restrictions on data, physics (PDE) or network architecture. [ABSTRACT FROM AUTHOR]

Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for such problems. Conventionally, SBP operators are tailored to the assumption that polynomials accurately approximate the solution, and SBP operators should thus be exact for them. However, this assumption falls short for a range of problems for which other approximation spaces are better suited. We recently addressed this issue and developed a theory for first-derivative SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the innovation of FSBP operators to accommodate second derivatives. The developed second-derivative FSBP operators maintain the desired mimetic properties of existing polynomial SBP operators while allowing for greater flexibility by being applicable to a broader range of function spaces. We establish the existence of these operators and detail a straightforward methodology for constructing them. By exploring various function spaces, including trigonometric, exponential, and radial basis functions, we illustrate the versatility of our approach. The work presented here opens up possibilities for using second-derivative SBP operators based on suitable function spaces, paving the way for a wide range of applications in the future. • We introduce second-derivative summation-by-parts operators for general (non-polynomial) function spaces. • We prove their existence and mimetic properties. • We present a straightforward construction procedure for general multi-dimensional domains/elements/blocks. [ABSTRACT FROM AUTHOR]

In many Bayesian inverse problems the goal is to recover a spatially varying random field. Such problems are often computationally challenging especially when the forward model is governed by complex partial differential equations (PDEs). The challenge is particularly severe when the spatial domain is large and the unknown random field needs to be represented by a high-dimensional parameter. In this paper, we present a domain-decomposed method to attack the dimensionality issue and the method decomposes the spatial domain and the parameter domain simultaneously. On each subdomain, a local Karhunen-Loève (KL) expansion is constructed, and a local inversion problem is solved independently in a parallel manner, and more importantly, in a lower-dimensional space. After local posterior samples are generated through conducting Markov chain Monte Carlo (MCMC) simulations on subdomains, a novel projection procedure is developed to effectively reconstruct the global field. In addition, the domain decomposition interface conditions are dealt with an adaptive Gaussian process-based fitting strategy. Numerical examples are provided to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

The lacking of analytic solutions of diverse partial differential equations (PDEs) gives birth to a series of computational techniques for numerical solutions. Although numerous latest advances are accomplished in developing neural operators, a kind of neural-network-based PDE solver, these solvers become less accurate and explainable while learning long-term behaviors of non-linear PDE families. In this paper, we propose the Koopman neural operator (KNO), a new neural operator, to overcome these challenges. With the same objective of learning an infinite-dimensional mapping between Banach spaces that serves as the solution operator of the target PDE family, our approach differs from existing models by formulating a non-linear dynamic system of equation solution. By approximating the Koopman operator, an infinite-dimensional operator governing all possible observations of the dynamic system, to act on the flow mapping of the dynamic system, we can equivalently learn the solution of a non-linear PDE family by solving simple linear prediction problems. We validate the KNO in mesh-independent, long-term, and zero-shot predictions on five representative PDEs (e.g., the Navier-Stokes equation and the Rayleigh-Bénard convection) and three real dynamic systems (e.g., global water vapor patterns and western boundary currents). In these experiments, the KNO exhibits notable advantages compared with previous state-of-the-art models, suggesting the potential of the KNO in supporting diverse science and engineering applications (e.g., PDE solving, turbulence modeling, and precipitation forecasting). • We propose a new neural operator model for partial differential equation (PDE) solving and realize an effective linear prediction of non-linear dynamics via Koopman operator. • Our approach achieves robust and accurate modeling of the long-term non-linear dynamics of PDE families or real systems. • Our model may serve as a basic unit for constructing new frameworks of PDE solving and dynamic system modeling. [ABSTRACT FROM AUTHOR]

Sparse-grid methods have recently gained interest in reducing the computational cost of solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid sparse-grid methods for the Vlasov–Poisson–Lenard–Bernstein (VPLB) model. This model has applications to plasma physics and is simulated in two reduced geometries: a 0 x 3 v space homogeneous geometry and a 1 x 3 v slab geometry. We use the discontinuous Galerkin (DG) method as a base discretization due to its high-order accuracy and ability to preserve important structural properties of partial differential equations. We utilize a multiwavelet basis expansion to determine the sparse-grid basis and the adaptive mesh criteria. We analyze the proposed sparse-grid methods on a suite of three test problems by computing the savings afforded by sparse-grids in comparison to standard solutions of the DG method. The results are obtained using the adaptive sparse-grid discretization library ASGarD. [ABSTRACT FROM AUTHOR]

Nowadays numerical methods for handling partial differential equations is so vast that it is almost taken for granted. In this paper, we propose a discrete duality finite volume scheme with biquadratic elements and its application in solving two-dimensional elliptic equation on quadrilateral meshes. To test the robustness and efficiency of this method, we investigate several examples from physical and engineering fields with different parameters and coefficients. Numerical experiments, encompassing linear and nonlinear elliptic equations with constant or variable coefficients, reveal that the new method achieves optimal convergence in the continuous H 1 -norm, with a 2nd-order convergence, and in the L 2 -norm, with a 3rd-order convergence. Moreover, comparison with the existing biquadratic element finite volume scheme shows that the above method has an advantage of accuracy in the discrete L ∞ -norm on several distorted meshes. [ABSTRACT FROM AUTHOR]

This paper investigates an efficient exponential integrator generalized multiscale finite element method for solving a class of time-evolving partial differential equations in bounded domains. The proposed method first performs the spatial discretization of the model problem using constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). This approach consists of two stages. First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. The multiscale basis functions are obtained in the second stage using the auxiliary space by solving local energy minimization problems over the oversampling domains. The basis functions have exponential decay outside the corresponding local oversampling regions. We shall consider the first and second-order explicit exponential Runge-Kutta approach for temporal discretization and to build a fully discrete numerical solution. The exponential integration strategy for the time variable allows us to take full advantage of the CEM-GMsFEM as it enables larger time steps due to its stability properties. We derive the error estimates in the energy norm under the regularity assumption. Finally, we will provide some numerical experiments to sustain the efficiency of the proposed method. • Multiscale reduction methods are efficient and accurate to solve problems with high-contrast media. • The presence of high-contrast coefficients reduces the stability region of time discretization. • Implicit methods must solve nonlinear equations, which can be a bottleneck computation. • Exponential integrators are alternative techniques and use robust time-stepping. [ABSTRACT FROM AUTHOR]

Pseudo-Hamiltonian neural networks (PHNN) were recently introduced for learning dynamical systems that can be modelled by ordinary differential equations. In this paper, we extend the method to partial differential equations. The resulting model is comprised of up to three neural networks, modelling terms representing conservation, dissipation and external forces, and discrete convolution operators that can either be learned or be given as input. We demonstrate numerically the superior performance of PHNN compared to a baseline model that models the full dynamics by a single neural network. Moreover, since the PHNN model consists of three parts with different physical interpretations, these can be studied separately to gain insight into the system, and the learned model is applicable also if external forces are removed or changed. • We introduce pseudo-Hamiltonian neural networks for partial differential equations. • The models can be separated into different parts, with physical interpretations. • Models learned on a disturbed system can predict future states without disturbances. • Assumptions on geometric structures can be imposed on the models. • Our new models consistently outperform the baseline neural network model. [ABSTRACT FROM AUTHOR]

This paper explores the feasibility of quantum simulation for partial differential equations (PDEs) with physical boundary or interface conditions. Semi-discretization of such problems does not necessarily yield Hamiltonian dynamics and even alters the Hamiltonian structure of the dynamics when boundary and interface conditions are included. This seemingly intractable issue can be resolved by using a recently introduced Schrödingerisation method [1,2] – it converts any linear PDEs and ODEs with non-Hermitian dynamics to a system of Schrödinger equations, via the so-called warped phase transformation that maps the equation into one higher dimension. We implement this method for several typical problems, including the linear convection equation with inflow boundary conditions and the heat equation with Dirichlet and Neumann boundary conditions. For interface problems we study the (parabolic) Stefan problem, linear convection, and linear Liouville equations with discontinuous and even measure-valued coefficients. We perform numerical experiments to demonstrate the validity of this approach, which helps to bridge the gap between available quantum algorithms and computational models for classical and quantum dynamics with boundary and interface conditions. • The first quantum simulation algorithms for PDEs with physical boundary conditions. • The first quantum simulation algorithms for PDEs incorporating physical interface conditions. • The first quantum algorithm for partial transmission and reflections from geometric optics with interfaces. [ABSTRACT FROM AUTHOR]

In this paper a new method for solving complex estimation problems for systems governed by partial differential equations and characterized by non-uniqueness (or equifinality) is presented. For such problems, the usefulness of the MAP estimates is doubtful, especially if the available prior information is very limited or not plausible in general. Thus, the posterior mean or median estimates would be more appropriate, which implies that the Bayesian approach has to be preferred. Since the direct use of the Bayesian approach involving computationally expensive models is not feasible a hybrid method which combines the Bayesian and variational elements in a unique way is suggested. The main idea of the method is to find one particular posterior mode out of a possibly infinite set, which would have the global properties (its first moments, for example) consistent with the available prior information and close to those of the posterior mean. The method is implemented in the form of a cyclic algorithm including the Bayesian estimation part for the lumped global (hidden, latent) variables describing the moments of the spatially distributed or time-dependent variables and the variational estimation part for the invariant 'shape' functions. The method has been numerically validated for two different applications: one including the Saint-Venant hydraulic model, and another one including the nonlinear convection-diffusion transport model. The results for the former are reported in a separate paper (reference provided), for the latter - in the present paper. These results confirm the expected properties of the suggested algorithm, such as improved robustness and accuracy, in comparison to the MAP estimator represented by the variational estimation algorithm. • We consider composite estimation problems for distributed parameter systems. • These problems are often characterized by non-uniqueness (equifinality). • We present a new Bayesian Variational Cyclic method designed for solving such problems. • The method is validated using the convection-diffusion and Saint-Venant models. • It shows improved robustness and accuracy in comparison to variational estimators. [ABSTRACT FROM AUTHOR]

In this paper, we develop algorithms to overcome the curse of dimensionality in non-convex state-dependent Hamilton-Jacobi partial differential equations (HJ PDEs) arising from optimal control and differential game problems. The subproblems are independent and they can be implemented in an embarrassingly parallel fashion. This is ideal for perfect scaling in parallel computing. The algorithm is proposed to overcome the curse of dimensionality [1,2] when solving HJ PDE. The major contribution of the paper is to change either the solving of a PDE problem or an optimization problem over a space of curves to an optimization problem of a single vector, which goes beyond the work of [40]. We extend the method in [7,9,15] , and conjecture a (Lax-type) minimization principle to solve state-dependent HJ PDE when the Hamiltonian is convex, as well as a (Hopf-type) maximization principle to solve state-dependent HJ PDE when the Hamiltonian is non-convex , as a generalization of the well-known Hopf formula in [18,25,50]. We showed the validity of the formula under restricted assumption for the sake of completeness, and would like to bring our readers to [62] which validates our conjectures in a more general setting. We conjectured the weakest assumption of our formula to hold is a pseudoconvexity assumption similar to one stated in [50]. Our method is expected to have application in control theory, differential game problems and elsewhere. [ABSTRACT FROM AUTHOR]

Abstract In the context of numerical solution of PDEs, dynamic mesh redistribution methods (r-adaptive methods) are an important procedure for increasing the resolution in regions of interest, without modifying the connectivity of the mesh. Key to the success of these methods is that the mesh should be sufficiently refined (locally) and flexible in order to resolve evolving solution features, but at the same time not introduce errors through skewness and lack of regularity. Some state-of-the-art methods are bottom-up in that they attempt to prescribe both the local cell size and the alignment to features of the solution. However, the resulting problem is overdetermined, necessitating a compromise between these conflicting requirements. An alternative approach, described in this paper, is to prescribe only the local cell size and augment this an optimal transport condition to provide global regularity. This leads to a robust and flexible algorithm for generating meshes fitted to an evolving solution, with minimal need for tuning parameters. Of particular interest for geophysical modelling are meshes constructed on the surface of the sphere. The purpose of this paper is to demonstrate that meshes generated on the sphere using this optimal transport approach have good a-priori regularity and that the meshes produced are naturally aligned to various simple features. It is further shown that the sphere's intrinsic curvature leads to more regular meshes than the plane. In addition to these general results, we provide a wide range of examples relevant to practical applications, to showcase the behaviour of optimally transported meshes on the sphere. These range from axisymmetric cases that can be solved analytically to more general examples that are tackled numerically. Evaluation of the singular values and singular vectors of the mesh transformation provides a quantitative measure of the mesh anisotropy, and this is shown to match analytic predictions. Graphical abstract [ABSTRACT FROM AUTHOR]

We are concerned with a novel Bayesian statistical framework for the characterization of natural subsurface formations, a very challenging task. Because of the large dimension of the stochastic space of the prior distribution in the framework, typically a dimensional reduction method, such as a Karhunen-Leove expansion (KLE), needs to be applied to the prior distribution to make the characterization computationally tractable. Due to the large variability of properties of subsurface formations (such as permeability and porosity) it may be of value to localize the sampling strategy so that it can better adapt to large local variability of rock properties. In this paper, we introduce the concept of multiscale sampling to localize the search in the stochastic space. We combine the simplicity of a preconditioned Markov Chain Monte Carlo method with a new algorithm to decompose the stochastic space into orthogonal subspaces, through a one-to-one mapping of the subspaces to subdomains of a non-overlapping domain decomposition of the region of interest. The localization of the search is performed by a multiscale blocking strategy within Gibbs sampling: we apply a KL expansion locally, at the subdomain level. Within each subdomain, blocking is applied again, for the sampling of the KLE random coefficients. The effectiveness of the proposed framework is tested in the solution of inverse problems related to elliptic partial differential equations arising in porous media flows. We use multi-chain studies in a multi-GPU cluster to show that the new algorithm clearly improves the convergence rate of the preconditioned MCMC method. Moreover, we illustrate the importance of a few conditioning points to further improve the convergence of the proposed method. [ABSTRACT FROM AUTHOR]

Domain decomposition methods are successful and highly parallel scalable iterative solution methods for discretized partial differential equations. Nevertheless, for many classes of problems, for example, elliptic partial differential equations with arbitrary coefficient distributions, adaptive coarse spaces are necessary to obtain robustness or, in other words, to guarantee a reliable and fast convergence. Adaptive coarse spaces are usually computed by solving many localized eigenvalue problems related to edges or faces of the domain decomposition. This results in a computationally expensive setup of the domain decomposition preconditioner or system operator. In earlier work, to which the authors have contributed, a deep learning based classification model was used to classify the critical edges or faces where eigenvalue problems have to be solved. In the present paper, we suggest to directly learn the adaptive constraints using a deep feedforward neural network regression model and thus completely skip the computationally most expensive part of the setup, i.e., the solution of local eigenvalue problems. We consider a specific adaptive FETI-DP (Finite Tearing and Interconnecting – Dual Primal) approach and concentrate on stationary diffusion problems in two dimensions with arbitrary coefficient functions with large jumps. As an input for the neural network, we use an image representation of the coefficient function which resolves the structure of the coefficient distribution but is not necessarily identical to the discretization of the partial differential equation. Therefore, our approach is independent of the finite element mesh and can, in principle, be easily extended to other adaptive coarse spaces, problems, and domain decomposition methods. We show the robustness of our method for different problems and the generalization property of our trained neural networks by considering different coefficient distributions not contained in the training set. We also combine the learned constraints with computationally cheap frugal constraints to further improve our approach. • For the first time, a neural network regression model is presented which predicts robust coarse constraints for FETI-DP. • The training data is obtained by computing provably robust coarse constraints for heterogeneous diffusion problems. • Our approach replaces the adaptive computation of coarse constraints by the fast evaluation of the regression model. • The method is mesh-independent and designed for regular and irregular domain decompositions and thus for a broad usability. • Several numerical results for complex and realistic stationary diffusion problems are presented. [ABSTRACT FROM AUTHOR]

Solutions of certain partial differential equations (PDEs) are often represented by the steepest descent curves of corresponding functionals. Minimizing movement scheme was developed in order to study such curves in metric spaces. Especially, Jordan-Kinderlehrer-Otto studied the Fokker-Planck equation in this way with respect to the Wasserstein metric space. In this paper, we propose a deep learning-based minimizing movement scheme for approximating the solutions of PDEs. The proposed method is highly scalable for high-dimensional problems as it is free of mesh generation. We demonstrate through various kinds of numerical examples that the proposed method accurately approximates the solutions of PDEs by finding the steepest descent direction of a functional even in high dimensions. • We propose a deep learning-based minimizing movement scheme for approximating the solutions of PDEs. • The proposed method is highly scalable for high-dimensional problems as it is free of mesh generation. • We demonstrate that the proposed method accurately approximates the solutions of PDEs even in high dimensions. [ABSTRACT FROM AUTHOR]

Algebraic multigrid (AMG) is an effective iterative algorithm for solving large-scale linear systems. One challenge of constructing the AMG algorithm is the determination of the prolongation operator, which affects the convergence rate of AMG and is problem-dependent. In this paper, we propose a new Learning-based Local Weighted Least Squares (L-LWLS) method to construct the prolongation operator of AMG. Specifically, we construct the prolongation operator by solving the LWLS model with learned spatially-varying weights. We use the gradient descent algorithm to optimize the model with a learned initialization of the solution. Then the constructed prolongation operator is further corrected by a learned correction function to improve the convergence rate of AMG. We conduct experiments on solving graph Laplacian linear systems, diffusion partial differential equations, and Helmholtz equations. Experiments show that the proposed method can construct a better prolongation operator leading to a faster convergence rate than the compared methods, including the classical AMG, the smoothed aggregation AMG, the bootstrap AMG, and the learning-based AMG method. The results show that the proposed method can generalize well to different parameter distributions and problem sizes, i.e., the number of variables in the linear system. • We proposed a learning-based method L-LWLS to optimize the prolongation operator in AMG. • We propose to learn spatially-varying weights in LWLS and learn to correct the LWLS solution. • The proposed method accelerates the convergence rate of AMG for parameterized PDEs. • The proposed method can generalize to different parameter distributions and problem sizes. [ABSTRACT FROM AUTHOR]

This paper proposes a novel computational modeling approach to investigate the fluid-structure interactions with moving contact lines. By embracing the generalized Onsager principle, a coupled hydrodynamics and phase field system is introduced that can describe the fluid-structure interactions with moving contact lines in a thermodynamically consistent manner. The resulting partial differential equation (PDE) model consists of the Navier-Stokes equation and a nonlinear Allen-Cahn type equation. Volume conservation is enforced through an additional penalty term. A fully-discrete structure-preserving numerical scheme is proposed by combining several techniques to solve this coupled PDE system effectively and accurately. For the temporal discretization, we utilize the supplementary variable method for preserving the thermodynamic structure and the projection approach for reducing the problem size. Then, we use the finite difference method on the staggered grid for spatial discretization. Furthermore, we have rigorously proved that the proposed numerical scheme based on the second-order backward difference formula respects the original energy stability, i.e., the scheme is energy stable. Additionally, with the aid of the supplementary variable method, the resultant scheme can be transformed into a constrained optimization problem, where the solutions of the supplementary variables are the arguments of the objective function that reaches the optimality. Then the augmented Lagrangian method is introduced in part to bring robustness and efficiency to solving such a constrained optimization problem. Finally, various numerical simulations verify the model's capability and demonstrate the scheme's effectiveness, accuracy, and stability. • We propose a novel model for fluid-structure interactions with moving contact lines. • Our PDE models embrace the generalized Onsager principle for thermodynamic consistency. • We introduce a supplementary variable to change the numerical scheme design into an optimization problem. • Contrary to other schemes, our numerical approach respects the original energy dissipation law. [ABSTRACT FROM AUTHOR]

In this paper we propose a new choice of poles to define reliable rational Krylov methods. These methods are used for approximating function of positive definite matrices. In particular, the fractional power and the fractional resolvent are considered because of their importance in the numerical solution of fractional partial differential equations. The numerical experiments on some fractional partial differential equation models confirm that the proposed approach is promising. [ABSTRACT FROM AUTHOR]

In this paper, we proposed two new types of edge multiscale methods motivated by [14] to solve Partial Differential Equations (PDEs) with high-contrast heterogeneous coefficients: Edge Spectral Multiscale Finite Element Method (ESMsFEM) and Wavelet-based Edge Multiscale Finite Element Method (WEMsFEM). Their convergence rates for elliptic problems with high-contrast heterogeneous coefficients are demonstrated in terms of the coarse mesh size H , the number of spectral basis functions and the level of the wavelet space ℓ , which are verified by extensive numerical tests. • We propose a new multiscale methods for problems with heterogeneous coefficients. • This new approach is very accurate and efficient. • We apply the approximation properties of the wavelets on the edges. • This new approach is very flexible and can be applied to other problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose a multi-symplectic quasi-interpolation method for solving multi-symplectic Hamiltonian partial differential equations. Based on the method of lines, we first discretize the multi-symplectic PDEs using quasi-interpolation method and then employ appropriate time integrators to obtain the full-discrete system. The local conservation properties including multi-symplectic conservation laws, energy conservation laws and momentum conservation laws are discussed in detail. For illustration, we provide two concrete examples: the nonlinear wave equation and the nonlinear Schrödinger equation. The salient feature of our multi-symplectic quasi-interpolation method is that it is valid both on uniform grids and nonuniform grids. The numerical results show the good accuracy and excellent conservation properties of the proposed method. • This is the first attempt to design multi-symplectic methods based on quasi-interpolation. • The multi-symplectic quasi-interpolation method has N + 1 local multi-symplectic conservation laws. • Our approach can easily deal with nonuniform grids, it is a meshless method. • The numerical results demonstrate that our method is efficient, easy to implement and also has good accuracy. [ABSTRACT FROM AUTHOR]

In this paper, the Monte Carlo Markov chain method for solving the modified anomalous fractional sub-diffusion equation is studied. Most of the previous methods are low in temporal and spatial accuracy order. Based on the idea of Monte Carlo Markov chain method and compact finite difference schemes, a probability model for solving the modified anomalous fractional sub-diffusion equation is established. Numerical examples are given to show the feasibility of the proposed scheme. Compared with the compact finite difference method, the present method is truly meshless and is easy to be implemented with high temporal and spatial accuracy order. And it is also applied to solve partial differential equation in irregular domains. • The Monte Carlo Markov chains method is developed to solve the modified anomalous fractional sub-diffusion equation. • The present method is truly meshless and is easy to be implemented with high temporal and spatial accuracy order. • The convergence orders are analyzed and discussed by calculating the numerical experiments. [ABSTRACT FROM AUTHOR]

• The discrete LS method is developed for two-dimensional time-fractional PDEs on irregular domains. • No mesh division is needed with easy implementation and high accuracy. • The LS method is easy to be extended to three-dimensional problems. • Regularization is applied to reduce the condition number of the coefficient matrix. In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using tensor product cubic spline basis functions defined on a background rectangular (interpolation) mesh, which leads to high spatial accuracy and straightforward implementation, and establishes a solid base for extending the computational framework to three-dimensional problems. A semi-implicit time-stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem. A key finding of our study is that the newly proposed mesh-free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero. Both methods produce a large constrained least-squares problem that must be solved at each time step in the advancement of the solution. We have found that regularization yields a relatively well-conditioned system that can be solved accurately using QR factorization. An extensive numerical investigation is performed to illustrate the effectiveness of the present methods, including the application of the new method to a coupled system of time-fractional partial differential equations having different fractional indices in different (irregularly shaped) regions of the solution domain. [ABSTRACT FROM AUTHOR]

This paper presents a simple meshfree approach, from the grid generation to the final solution, for the simulation of 3D problems geometrically defined by CAD. First, the domain grid for 3D problems is generated through a discrete searching algorithm. Non-Uniform Rational B-Splines (NURBS) are employed, just as a choice among similar tools, to define the boundaries through geometrical control points obtained by the CAD program. A predefined regular grid of nodes, embedding the whole geometry, is then trimmed to follow the constructed boundaries. The spatial solution is performed by construction of shape functions satisfying the governing differential equation, through using exponential basis functions (EBFs). A straightforward strategy is proposed for choosing appropriate EBFs via their shape-parameters making the method efficient for solution of 3D problems. Several 3D Laplace and Helmholtz problems are solved and the results are compared with those of commercial programs to show the efficiency of the method. • Automatic/systematic determination of parameters in the exponential basis functions. • Kronecker-delta property of the shape-functions. • Reducing the computational effort by removing the pseudo inverse procedure compared to the former version. • The accuracy of the proposed method is verified through solving the Laplace and Helmholtz equations. • The domain grid for 3D problems is generated through a point-in-membership algorithm. [ABSTRACT FROM AUTHOR]

This paper is devoted to the derivation of computational methods for constructing partial differential equations from data. Following some recent works [7,14,15,20] , we propose a methodology based on symbolic calculus [8,9,13] , pseudospectral methods [2,3] and stochastic processes [6] , in order to determine non-constant coefficients of linear evolution Partial Differential Equations (PDEs), from a set of structured data constituted by solutions at given times and positions, of an unknown linear PDE. • Reconstruction of differential operator symbols from randomly perturbed data, using i) symbolic calculus, ii) pseudospectral methods, and iii) polynomial interpolation. • Numerical experiments. [ABSTRACT FROM AUTHOR]

• Multifidelity method that enables estimation of failure probabilities for expensive-to- evaluate models. • For turbulent jet application, it reduces the CPU time to compute the biasing density by 65%. • The new approach draws from information fusion, importance sampling and multifidelity modeling. • The fused estimator is optimal in that it has minimal variance among all possible combinations of the estimators. This paper develops a multifidelity method that enables estimation of failure probabilities for expensive-to-evaluate models via information fusion and importance sampling. The presented general fusion method combines multiple probability estimators with the goal of variance reduction. We use low-fidelity models to derive biasing densities for importance sampling and then fuse the importance sampling estimators such that the fused multifidelity estimator is unbiased and has mean-squared error lower than or equal to that of any of the importance sampling estimators alone. By fusing all available estimators, the method circumvents the challenging problem of selecting the best biasing density and using only that density for sampling. A rigorous analysis shows that the fused estimator is optimal in the sense that it has minimal variance amongst all possible combinations of the estimators. The asymptotic behavior of the proposed method is demonstrated on a convection-diffusion-reaction partial differential equation model for which 105 samples can be afforded. To illustrate the proposed method at scale, we consider a model of a free plane jet and quantify how uncertainties at the flow inlet propagate to a quantity of interest related to turbulent mixing. Compared to an importance sampling estimator that uses the high-fidelity model alone, our multifidelity estimator reduces the required CPU time by 65% while achieving a similar coefficient of variation. [ABSTRACT FROM AUTHOR]

The present paper develops two new techniques, namely additive correction multicloud (ACMC) and smoothed restriction multicloud (SRMC), for the efficient solution of systems of equations arising from Radial Basis Function-generated Finite Difference (RBF-FD) meshless discretizations of partial differential equations (PDEs). RBF-FD meshless methods employ arbitrary distributed nodes, without the need to generate a mesh, for the numerical solution of PDEs. The proposed techniques are specifically designed for the RBF-FD data structure and employ simple restriction and interpolation strategies in order to obtain a hierarchy of coarse-level node distributions and the corresponding correction equations. Both techniques are kept as simple as possible in terms of code implementation, which is an important feature of meshless methods. The techniques are verified on 2D and 3D Poisson equations, defined on non-trivial domains , showing very high benefits in terms of both time consumption and work to convergence when comparing the present techniques to the most common solver approaches. These benefits make the RBF-FD approach competitive with standard grid-based approaches when the number of nodes is very high, allowing large size problems to be tackled by the RBF-FD method. • Multilevel principles for convergence acceleration are applied to RBF-FD meshless discretizations. • Two simple multicloud techniques have been developed and successfully applied. • Code implementation is kept as simple as possible. • Convergence work and computing time are considerably reduced (up to 10× and 20×, respectively). • Possibility to face very large size problems with the RBF-FD meshless method. [ABSTRACT FROM AUTHOR]

Partial discharges and the electrical treeing represent one of key aspects for the ageing of the insulating parts of electrical components. To predict the end of life of dielectric systems several different modelling approaches have been developed. One of the most promising is the modelling using first principles and a set of partial differential equations. In this field, one of the toughest challenges is to cope with the very different scales of the treeing (each branch is a few micrometers wide) and of electrical components (raging from a few centimetres to a few meters). In this paper we propose an approach, based on the finite element and finite volume methods, which uses two completely independent grids to describe internal defects and electrical components. Some interpolation algorithms have been developed to couple the solution on the two grids. We have also tested the accuracy of our approach by comparing it with previously published methods that use only one grid. • Partial discharges simulation using plasma physics for challenging geometries. • Efficient treatment of tiny internal defects in electric equipments. • Grid decoupling between a gaseous defect and a dielectric bulk. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel methodology for automatic adaptive weighting of Bayesian Physics-Informed Neural Networks (BPINNs), and we demonstrate that this makes it possible to robustly address multi-objective and multiscale problems. BPINNs are a popular framework for data assimilation, combining the constraints of Uncertainty Quantification (UQ) and Partial Differential Equation (PDE). The relative weights of the BPINN target distribution terms are directly related to the inherent uncertainty in the respective learning tasks. Yet, they are usually manually set a-priori, that can lead to pathological behavior, stability concerns, and to conflicts between tasks which are obstacles that have deterred the use of BPINNs for inverse problems with multiscale dynamics. The present weighting strategy automatically tunes the weights by considering the multitask nature of target posterior distribution. We show that this remedies the failure modes of BPINNs and provides efficient exploration of the optimal Pareto front. This leads to better convergence and stability of BPINN training while reducing sampling bias. The determined weights moreover carry information about task uncertainties, reflecting noise levels in the data and adequacy of the PDE model. We demonstrate this in numerical experiments in Sobolev training, and compare them to analytically ε -optimal baseline, and in a multiscale Lotka-Volterra inverse problem. We eventually apply this framework to an inpainting task and an inverse problem, involving latent field recovery for incompressible flow in complex geometries. • Adaptive weighting enables robust uncertainty quantification in multitask Bayesian inference. • It avoids Bayesian PINNs training failure and provides unbiased uncertainty estimates. • It avoids manual noise parameter tuning and allows solving multi-scale problems. [ABSTRACT FROM AUTHOR]

This article proposes an efficient numerical method for solving nonlinear partial differential equations (PDEs) based on sparse Gaussian processes (SGPs). Gaussian processes (GPs) have been extensively studied for solving PDEs by formulating the problem of finding a reproducing kernel Hilbert space (RKHS) to approximate a PDE solution. The approximated solution lies in the span of base functions generated by evaluating derivatives of different orders of kernels at sample points. However, the RKHS specified by GPs can result in an expensive computational burden due to the cubic computation order of the matrix inverse. We conjecture that a solution exists on a "condensed" subspace that can achieve similar approximation performance, and we propose a SGP-based method to reformulate the optimization problem in the "condensed" subspace. This significantly reduces the computation burden while retaining desirable accuracy. The paper rigorously formulates this problem and provides error analysis and numerical experiments to demonstrate the effectiveness of this method. The numerical experiments show that the SGP method uses fewer than half the uniform samples as inducing points and achieves comparable accuracy to the GP method using the same number of uniform samples, resulting in a significant reduction in computational cost. Our contributions include formulating the nonlinear PDE problem as an optimization problem on a "condensed" subspace of RKHS using SGP, as well as providing an existence proof and rigorous error analysis. Furthermore, our method can be viewed as an extension of the GP method to account for general positive semi-definite kernels. • We propose a new algorithm for solving nonlinear partial differential equations based on sparse Gaussian processes. • Our algorithms base on theories of kernel methods and sparse Gaussian processes. • Our algorithm is meshless and flexible to the shape of domains. • The sparse Gaussian process method reduces computational complexity by using low-rank kernels. [ABSTRACT FROM AUTHOR]

Physics-informed Neural Network (PINN) is a promising tool that has been applied in a variety of physical phenomena described by partial differential equations (PDE). However, it has been observed that PINNs are difficult to train in certain "stiff" problems, which include various nonlinear hyperbolic PDEs that display shocks in their solutions. Recent studies added a diffusion term to the PDE, and an artificial viscosity (AV) value was manually tuned to allow PINNs to solve these problems. In this paper, we propose three approaches to address this problem, none of which rely on an a priori definition of the artificial viscosity value. The first method learns a global AV value, whereas the other two learn localized AV values around the shocks, by means of a parametrized AV map or a residual-based AV map. We applied the proposed methods to the inviscid Burgers equation and the Buckley-Leverett equation, the latter being a classical problem in Petroleum Engineering. The results show that the proposed methods are able to learn both a small AV value and the accurate shock location and improve the approximation error over a nonadaptive global AV alternative method. • Methods to add adaptive artificial viscosity to train PINNs to solve hyperbolic PDEs with shocks. • Adaptive methods accurately learn both the value and location of the artificial viscosity. • Successfully solved the Inviscid Burgers and Buckley-Leverett equations. • Additional computation cost over the baseline PINN is small. [ABSTRACT FROM AUTHOR]

This paper presents a fast algorithm for computing transport properties of two-dimensional Dirac operators with linear domain walls, which model the macroscopic behavior of the robust and asymmetric transport observed at an interface separating two two-dimensional topological insulators. Our method is based on reformulating the partial differential equation as a corresponding volume integral equation, which we solve via a spectral discretization scheme. We demonstrate the accuracy of our method by confirming the quantization of an appropriate interface conductivity modeling transport asymmetry along the interface, and moreover, confirm that this quantity is immune to local perturbations. We also compute the far-field scattering matrix generated by such perturbations and verify that while asymmetric transport is topologically protected the absence of back-scattering is not. • A fast and accurate algorithm to compute transport properties of 2D Dirac model. • Interface conductivity is quantized and immune to local perturbations. • The asymmetric transport is shown to be topologically protected. • A merging-and-recovering scheme is developed to reduce computational cost. [ABSTRACT FROM AUTHOR]

The goal of this paper is to develop a novel semi-implicit spectral deferred correction (SDC) time marching method. The method can be used in a large class of problems, especially for highly nonlinear ordinary differential equations (ODEs) without easily separating of stiff and non-stiff components, which is more general and efficient comparing with traditional semi-implicit SDC methods. The proposed semi-implicit SDC method is based on low order time integration methods and corrected iteratively. The order of accuracy is increased for each additional iteration. And we also explore its local truncation error analytically. This SDC method is intended to be combined with the method of lines, which provides a flexible framework to develop high order semi-implicit time marching methods for nonlinear partial differential equations (PDEs). In this paper we mainly focus on the applications of the nonlinear PDEs with higher order spatial derivatives, e.g. convection diffusion equation, the surface diffusion and Willmore flow of graphs, the Cahn–Hilliard equation, the Cahn–Hilliard–Brinkman system and the phase field crystal equation. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the fully discrete schemes are all high order accurate in both space and time, and stable numerically with the time step proportional to the spatial mesh size. Numerical experiments are carried out to illustrate the accuracy and capability of the proposed semi-implicit SDC method. [ABSTRACT FROM AUTHOR]

Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r = 0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy. [ABSTRACT FROM AUTHOR]

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained. • The Krylov implicit integration factor discontinuous Galerkin methods were first designed on sparse grids. • The new methods can simulate high spatial dimensional reaction-diffusion equations efficiently. • Both theoretical analysis and numerical experiments were performed to study the new methods. [ABSTRACT FROM AUTHOR]

Fig. 1 A one-form (vector field) given as edge values on a 128 × 128 grid is decomposed into a coarse one-form component u 1 (on a 2 × 2 grid) and a series of wavelet coefficients for a six-level hierarchy adapted to the one-form Laplace operator; the relative (Dirichlet) energy content of each level is indicated as well. Fig. 1 • We propose the first construction of operator-adapted wavelets for finite-element differential forms. • The core of our construction relies on refinable and subdivision-based high-order bases of arbitrary differential forms. • Since our approach applies to arbitrary differential forms, we can derive scalar-valued as well as vector-valued wavelets. • The resulting wavelets are scale-orthogonal wrt a linear operator L , leading to a Galerkin discretization that is block-diagonal, with well-conditioned blocks. • Our construction accommodates linear differential constraints such as divergence-freeness, and has direct applications such as coarse-graining and model reduction of PDEs. We introduce in this paper an operator-adapted multiresolution analysis for finite-element differential forms. From a given continuous, linear, bijective, and self-adjoint positive-definite operator L , a hierarchy of basis functions and associated wavelets for discrete differential forms is constructed in a fine-to-coarse fashion and in quasilinear time. The resulting wavelets are L -orthogonal across all scales, and can be used to derive a Galerkin discretization of the operator such that its stiffness matrix becomes block-diagonal, with uniformly well-conditioned and sparse blocks. Because our approach applies to arbitrary differential p -forms, we can derive both scalar-valued and vector-valued wavelets block-diagonalizing a prescribed operator. We also discuss the generality of the construction by pointing out that it applies to various types of computational grids, offers arbitrary smoothness orders of basis functions and wavelets, and can accommodate linear differential constraints such as divergence-freeness. Finally, we demonstrate the benefits of the corresponding operator-adapted multiresolution decomposition for coarse-graining and model reduction of linear and non-linear partial differential equations. [ABSTRACT FROM AUTHOR]

Abstract Constructing stable difference schemes on complex geometries is an arduous task. Even fairly simple partial differential equations end up very convoluted in their discretized form, making them difficult to implement and manage. Spatial discretizations using so called summation-by-parts operators have mitigated this issue to some extent, particularly on rectangular domains, making it possible to formulate stable discretizations in a compact and understandable manner. However, the simplicity of these formulations is lost for curvilinear grids, where the standard procedure is to transform the grid to a rectangular one, and change the structure of the original equation. In this paper we reinterpret the grid transformation as a transformation of the summation-by-parts operators. This results in operators acting directly on the curvilinear grid. Together with previous developments in the field of nonconforming grid couplings we can formulate simple, implementable, and provably stable schemes on general nonconforming curvilinear grids. The theory is applicable to methods on summation-by-parts form, including finite differences, discontinuous Galerkin spectral element, finite volume, and flux reconstruction methods. Time dependent advection–diffusion simulations corroborate the theoretical development. [ABSTRACT FROM AUTHOR]

Abstract Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to derive the relations between the involved physical quantities of interest. A different approach is to measure the quantities of interest and use deep learning to reverse engineer the PDEs which are describing the physical process. In this paper we use machine learning, and deep learning in particular, to discover PDEs hidden in complex data sets from measurement data. We include examples of data from a known model problem, and real data from weather station measurements. We show how necessary transformations of the input data amounts to coordinate transformations in the discovered PDE, and we elaborate on feature and model selection. It is shown that the dynamics of a non-linear, second order PDE can be accurately described by an ordinary differential equation which is automatically discovered by our deep learning algorithm. Even more interestingly, we show that similar results apply in the context of more complex simulations of the Swedish temperature distribution. [ABSTRACT FROM AUTHOR]

Abstract In many spinning processes, as for example in dry spinning, solvent evaporates out of the spun jets and leads to thinning and solidification of the produced fibers. Such production processes are significantly driven by the interaction of the fibers with the surrounding airflow. Faced with industrial applications producing up to several hundred fibers simultaneously, the direct numerical simulation of the three-dimensional multiphase, multiscale problem is computationally extremely demanding and thus in general not possible. In this paper, we hence propose a dimensionally reduced, efficiently evaluable fiber model that enables the realization of fiber-air interactions in a two-way coupling with airflow computations. For viscous dry spinning of an uni-axial two-phase flow, we deduce one-dimensional equations for fiber velocity and stress from cross-sectional averaging and combine them with two-dimensional advection-diffusion equations for polymer mass fraction and temperature revealing the radial effects that are observably present in experiments. For the numerical treatment of the resulting parametric boundary value problem composed of one-dimensional ordinary differential equations and two-dimensional partial differential equations we develop an iterative coupling algorithm. The solution of the advection-diffusion equations is implicitly given in terms of Green's functions and leads for the surface values to Volterra integral equations of second kind with singular kernel, which we can solve very efficiently by the product integration method. For the ordinary differential equations a suitable collocation-continuation procedure is presented. Compared with the referential solution of a three-dimensional setting, the numerical results are very convincing. They provide a good approximation while drastically reducing the computational time. This efficiency allows the two-way coupled simulation of industrial dry spinning in airflows for which we present results for the first time in literature. [ABSTRACT FROM AUTHOR]

Abstract The closest point method (Ruuth and Merriman (2008) [32]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao (2009) [37]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection–diffusion equations and Cahn–Hilliard equations that are strongly coupled to the velocity of the surface are also presented. Highlights • In this paper, a novel least-squares approach to stabilize the RBF closest point method is proposed. • The use of RBF-FD allows the construction of higher-order accurate results by increasing the number of points in the stencil. • Convergence is also shown for irregular types of computational tubes in a neighborhood of the surface. • A coupled algorithm for the solution of PDEs on moving surfaces is introduced. • Convergence tests and examples of reaction–diffusion and Cahn–Hilliard systems on moving surfaces are presented. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a generalised perturbation form of the nonhydrostatic partial differential equations (PDEs) that govern dynamics of all-scale global atmospheric flows. There can be many alternative perturbation forms for any given system of the governing PDEs, depending on the assumed ambient state about which perturbations are taken and on subject preferences in the numerical model design. All such forms are mathematically equivalent, yet they have different implications for the design and accuracy of effective semi-implicit numerical integrators of the governing PDEs. Practical and relevant arguments are presented in favour of perturbation forms that maximise the degree of implicitness of the associated integrators. Two optional forms are implemented in the high-performance finite-volume module (IFS-FVM) for simulating global all-scale atmospheric flows Smolarkiewicz et al. (2016) [32]. Their relative performance is verified with a class of ambient states of reduced complexity. A series of numerical simulations of the planetary baroclinic instability assumes geostrophically balanced zonally uniform ambient flows with significant meridional and vertical shear and exemplifies accuracy gains enabled by the generalised perturbation approach. The newly developed semi-implicit integrators show the potential for numerically accurate separation of a background state from finite-amplitude perturbations of the global atmosphere and provide a base for further development. Highlights • New perturbation equations for of all-scale global atmospheric dynamics are derived. • Novel semi-implicit nonoscillatory forward-in-time solvers are developed for the derived PDEs. • Simulations of global baroclinic instability verify the new equations and their solvers. • Simulations show improved accuracy of the new solvers for extended range forecasting. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a data-driven compressive sensing method for the effective and sparse representation of solutions to a stochastic system, in which a problem-dependent basis is constructed and used to exploit the sparse patterns. Given what appears to be a highly incomplete set of sample observations, the essential ingredients of the proposed method involve (i) estimating the covariance function from low-fidelity simulations, (ii) extracting the most dominant energetic modes from the Karhunen-Loève analysis, and (iii) solving the basis pursuit problem associated with data-driven functions for high-fidelity models. Compared with other conventional compressive sensing methods that use a problem-independent basis, our approach can significantly increase the sparsity of expansion coefficients in many situations. When applied to uncertainty quantification problems such as partial differential equations with random input data, numerical experiments are carried out to show the effectiveness of our data-driven method in recovering the sparse solution globally over the physical domain. [ABSTRACT FROM AUTHOR]

Abstract The current paper presents a large time step wave adding scheme (hereafter LTS-WA) originated in LeVeque's large time Godunov scheme, i.e. constructing numerical schemes via adding strong waves of discontinuity decomposition. Compared with the original scheme, we use a different strategy for wave adding and extend it to multi-dimensional cases. For rarefaction waves, we propose a grid cell decomposition method which can automatically satisfy the entropy condition and avoid nonphysical solutions. We give detailed formulae of the scheme, and make numerical experiments using scalar equations and Euler equations in one and multi-dimensional cases. Numerical results show that, besides the advantage of large time step, the new scheme has a lower numerical dissipation and a higher resolution of shocks and contact discontinuities with the increase of CFL number in certain range. Highlights • Large time step (CFL > 1). • High resolution. • Increasing resolution with CFL number. • Zero error at integer CFL number in linear case. [ABSTRACT FROM AUTHOR]

This paper presents a new algorithm KIOPS for computing linear combinations of φ -functions that appear in exponential integrators. This algorithm is suitable for large-scale problems in computational physics where little or no information about the spectrum or norm of the Jacobian matrix is known a priori . We first show that such problems can be solved efficiently by computing a single exponential of a modified matrix. Then our approach is to compute an appropriate basis for the Krylov subspace using the incomplete orthogonalization procedure and project the matrix exponential on this subspace. We also present a novel adaptive procedure that significantly reduces the computational complexity of exponential integrators. Our numerical experiments demonstrate that KIOPS outperforms the current state-of-the-art adaptive Krylov algorithm phipm . [ABSTRACT FROM AUTHOR]

Spectral methods are becoming increasingly prevalent in solving time-varying partial differential equations due to their fast convergence properties. However, they typically use regular computational meshes that do not account for spatially varying resolution requirements. This can significantly increase the overall grid density when resolution requirements vary sharply over the modelled domain. Moving mesh methods offer a remedy for this, by allowing the position of mesh nodes to adapt to the simulated model solution. In this paper, a mesh specification is presented that is based on a local measure of the spatial bandwidth of the model solution. This addresses the rate of decay of the model solution's frequency components by producing high-sampling rates when this decay is slow. The spatial bandwidth is computed using a combination of the original solution and its Riesz transformed counterparts. It is then integrated into a Fourier spectral moving mesh method, using the parabolic Monge–Ampère equation for mesh control. This method is used to solve a multidimensional version of the viscous Burgers equation, and a heterogeneous advection equation. The performance of bandwidth-based mesh adaptation is compared with arclength- and curvature-based adaptation, and against a static mesh. These numerical experiments show that the bandwidth-based approach produces superior convergence rates, and hence requires fewer mesh nodes for a given level of solution accuracy. [ABSTRACT FROM AUTHOR]

In the strong scaling limit, the performance of conventional spatial domain decomposition techniques for the parallel solution of PDEs saturates. When sub-domains become small, halo-communication and other overhead come to dominate. A potential path beyond this scaling limit is to introduce domain-decomposition in time, with one such popular approach being the Parareal algorithm which has received a lot of attention due to its generality and potential scalability. Low efficiency, particularly on convection dominated problems, has however limited the adoption of the method. In this paper we demonstrate trough large-scale numerical experiments that it is possible not only to obtain time-parallel speedup on the non-linear shallow water wave equation, but also that we may obtain parallel acceleration beyond what is possible using conventional spatial domain-decomposition techniques alone. Two factors were essential in achieving this. First, for Parareal to converge on the hyperbolic problem we used an approximate Riemann solver as the preconditioner. This preconditioner introduces only dissipative errors with respect to the 3rd order accurate WENO-RK discretization used to solve the PDE system. The preconditoner is comparatively expensive and convergence is slow unless time-subdomains are short. We therefore introduce a new scheduler that we denote Communication Aware Adaptive Parareal (CAAP). CAAP increases obtainable speed-up by minimizing the time-subdomain length without making communication of time-subdomains too costly whilst also adaptively overlapping consecutive cycles of Parareal so to mitigate the impact of a relatively expensive coarse operator. [ABSTRACT FROM AUTHOR]

Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman (2008) [17] ) is an embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF-FD). Unlike the orthogonal gradients method (Piret (2012) [22] ), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao (2009) [26] ). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method. [ABSTRACT FROM AUTHOR]

This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many applications due to their distinctive features. However, they are often deemed too costly because of the large degrees of freedom of the approximation space, which are the main bottleneck for simulations in high dimensions. In this paper, we develop sparse grid DG methods for elliptic equations with the aim of breaking the curse of dimensionality . Using a hierarchical basis representation, we construct a sparse finite element approximation space, reducing the degrees of freedom from the standard O ( h − d ) to O ( h − 1 | log 2 ⁡ h | d − 1 ) for d -dimensional problems, where h is the uniform mesh size in each dimension. Our method, based on the interior penalty (IP) DG framework, can achieve accuracy of O ( h k | log 2 ⁡ h | d − 1 ) in the energy norm, where k is the degree of polynomials used. Error estimates are provided and confirmed by numerical tests in multi-dimensions. [ABSTRACT FROM AUTHOR]

In this paper a new, simple and universal formulation of the HLLEM Riemann solver (RS) is proposed that works for general conservative and non-conservative systems of hyperbolic equations. For non-conservative PDE, a path-conservative formulation of the HLLEM RS is presented for the first time in this paper. The HLLEM Riemann solver is built on top of a novel and very robust path-conservative HLL method. It thus naturally inherits the positivity properties and the entropy enforcement of the underlying HLL scheme. However, with just the slight additional cost of evaluating eigenvectors and eigenvalues of intermediate characteristic fields, we can represent linearly degenerate intermediate waves with a minimum of smearing. For conservative systems, our paper provides the easiest and most seamless path for taking a pre-existing HLL RS and quickly and effortlessly converting it to a RS that provides improved results, comparable with those of an HLLC, HLLD, Osher or Roe-type RS. This is done with minimal additional computational complexity, making our variant of the HLLEM RS also a very fast RS that can accurately represent linearly degenerate discontinuities. Our present HLLEM RS also transparently extends these advantages to non-conservative systems. For shallow water-type systems, the resulting method is proven to be well-balanced. Several test problems are presented for shallow water-type equations and two-phase flow models, as well as for gas dynamics with real equation of state, magnetohydrodynamics (MHD & RMHD), and nonlinear elasticity. Since our new formulation accommodates multiple intermediate waves and has a broader applicability than the original HLLEM method, it could alternatively be called the HLLI Riemann solver, where the “I” stands for the intermediate characteristic fields that can be accounted for. [ABSTRACT FROM AUTHOR]