High Order Finite Difference WENO Methods with Unequal-Sized Sub-Stencils for the Degasperis-Procesi Type Equations

In this paper, we develop two finite difference weighted essentially non-oscillatory (WENO) schemes with unequal-sized sub-stencils for solving the Degasperis-Procesi (DP) and $\mu$-Degasperis-Procesi ($\mu$DP) equations, which contain nonlinear high order derivatives, and possibly peakon solutions or shock waves. By introducing auxiliary variable(s), we rewrite the DP equation as a hyperbolic-elliptic system, and the $\mu$DP equation as a first order system. Then we choose a linear finite difference scheme with suitable order of accuracy for the auxiliary variable(s), and two finite difference WENO schemes with unequal-sized sub-stencils for the primal variable. One WENO scheme uses one large stencil and several smaller stencils, and the other WENO scheme is based on the multi-resolution framework which uses a series of unequal-sized hierarchical central stencils. Comparing with the classical WENO scheme which uses several small stencils of the same size to make up a big stencil, both WENO schemes with unequal-sized sub-stencils are simple in the choice of the stencil and enjoy the freedom of arbitrary positive linear weights. Another advantage is that the final reconstructed polynomial on the target cell is a polynomial of the same degree as the polynomial over the big stencil, while the classical finite difference WENO reconstruction can only be obtained for specific points inside the target interval. Numerical tests are provided to demonstrate the high order accuracy and non-oscillatory properties of the proposed schemes.     

A New Type of High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes

In this paper, a new type of third-order and fourth-order weighted essentially non-oscillatory (WENO) schemes is designed for simulating the Hamilton-Jacobi equations on triangular meshes. We design such schemes with the use of the nodal information defined on five unequal-sized spatial stencils, the application of monotone Hamiltonians as a building block, the artificial set of positive linear weights to make up high-order approximations in smooth regions simultaneously avoiding spurious oscillations nearby discontinuities of the derivatives of the solutions. The spatial reconstructions are convex combinations of the derivatives of a modified cubic/quartic polynomial defined on a big spatial stencil and four quadratic polynomials defined on small spatial stencils, and a third-order TVD Runge-Kutta method is used for the time discretization. The main advantages of these WENO schemes are their efficiency, simplicity, and can be easily implemented to higher dimensional unstructured meshes. Extensive numerical tests are performed to illustrate the good performance of such new WENO schemes.     

A New Approach of High Order Well-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms. In our earlier work [31–33], we designed high order well-balanced schemes to a class of hyperbolic systems with separable source terms. In this paper, we present a different approach to the same purpose: designing high order well-balanced finite volume weighted essentially non-oscillatory (WENO) schemes and RungeKutta discontinuous Galerkin (RKDG) finite element methods. We make the observation that the traditional RKDG methods are capable of maintaining certain steady states exactly, if a small modification on either the initial condition or the flux is provided. The computational cost to obtain such a well balanced RKDG method is basically the same as the traditional RKDG method. The same idea can be applied to the finite volume WENO schemes. We will first describe the algorithms and prove the well balanced property for the shallow water equations, and then show that the result can be generalized to a class of other balance laws. We perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions.     

High-Order Runge-Kutta Discontinuous Galerkin Methods with a New Type of Multi-Resolution WENO Limiters on Tetrahedral Meshes

In this paper, the second-order and third-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (WENO) limiters are proposed on tetrahedral meshes. The multi-resolution WENO limiter is an extension of a finite volume multi-resolution WENO scheme developed in [81], which serves as a limiter for RKDG methods on tetrahedral meshes. This new WENO limiter uses information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCF indicator [42], to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the second or third degree of the DG solution. The second-order and third-order RKDG methods with the associated multi-resolution WENO limiters are developed as examples for general high-order RKDG methods, which could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property near strong discontinuities by gradually degrading from the optimal order to the first order. The linear weights inside the procedure of the new multi-resolution WENO limiters can be set as any positive numbers on the condition that they sum to one. A series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell on tetrahedral meshes. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy on tetrahedral meshes. Such spatial reconstruction methodology improves the robustness in the simulation on the same compact spatial stencil of the original DG methods on tetrahedral meshes. Extensive one-dimensional (run as three-dimensional problems on tetrahedral meshes) and three-dimensional tests are performed to demonstrate the good performance of the RKDG methods with new multi-resolution WENO limiters.     

Adaptive Order WENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem

We present a new conservative semi-Lagrangian finite difference weighted essentially non-oscillatory scheme with adaptive order. This is an extension of the conservative semi-Lagrangian (SL) finite difference WENO scheme in [Qiu and Shu, JCP, 230 (4) (2011), pp. 863-889], in which linear weights in SL WENO framework were shown not to exist for variable coefficient problems. Hence, the order of accuracy is not optimal from reconstruction stencils. In this paper, we incorporate a recent WENO adaptive order (AO) technique [Balsara et al., JCP, 326 (2016), pp. 780-804] to the SL WENO framework. The new scheme can achieve an optimal high order of accuracy, while maintaining the properties of mass conservation and non-oscillatory capture of solutions from the original SL WENO. The positivity-preserving limiter is further applied to ensure the positivity of solutions. Finally, the scheme is applied to high dimensional problems by a fourth-order dimensional splitting. We demonstrate the effectiveness of the new scheme by extensive numerical tests on linear advection equations, the Vlasov-Poisson system, the guiding center Vlasov model as well as the incompressible Euler equations.     

A Sufficient and Necessary Condition of the Existence of WENO-Like Linear Combination for Finite Difference Schemes

In the finite difference WENO (weighted essentially non-oscillatory) method, the final scheme on the whole stencil was constructed by linear combinations of highest order accurate schemes on sub-stencils, all of which share the same total count of grid points. The linear combination method which the original WENO applied was generalized to arbitrary positive-integer-order derivative on an arbitrary (uniform or non-uniform) mesh, still applying finite difference method. The possibility of expressing the final scheme on the whole stencil as a linear combination of highest order accurate schemes on WENO-like sub-stencils was investigated. The main results include: (a) the highest order of accuracy a finite difference scheme can achieve and (b) a sufficient and necessary condition that the linear combination exists. This is a sufficient and necessary condition for all finite difference schemes in a set (rather than a specific finite difference scheme) to have WENO-like linear combinations. After the proofs of the results, some remarks on the WENO schemes and TENO (targeted essentially non-oscillatory) schemes were given.     

A Slope Constrained 4th Order Multi-Moment Finite Volume Method with WENO Limiter

This paper presents a new and better suited formulation to implement the limiting projection to high-order schemes that make use of high-order local reconstructions for hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-moment Constrained finite Volume with WENO limiter of 4th order) method, is an extension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative (gradient or slope) at the cell center as an additional constraint for the cell-wise local reconstruction. The gradient is computed from a limiting projection using the WENO (weighted essentially non-oscillatory) reconstruction that is built from the nodal values at 5 solution points within 3 neighboring cells. Different from other existing methods where only the cell-average value is used in the WENO reconstruction, the present method takes account of the solution structure within each mesh cell, and thus minimizes the stencil for reconstruction. The resulting scheme has 4th-order accuracy and is of significant advantage in algorithmic simplicity and computational efficiency. Numerical results of one and two dimensional benchmark tests for scalar and Euler conservation laws are shown to verify the accuracy and oscillation-less property of the scheme.     

Runge-Kutta Discontinuous Galerkin Method with a Simple and Compact Hermite WENO Limiter

In this paper, we propose a new type of weighted essentially non-oscillatory (WENO) limiter, which belongs to the class of Hermite WENO (HWENO) limiters, for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving hyperbolic conservation laws. This new HWENO limiter is a modification of the simple WENO limiter proposed recently by Zhong and Shu [29]. Both limiters use information of the DG solutions only from the target cell and its immediate neighboring cells, thus maintaining the original compactness of the DG scheme. The goal of both limiters is to obtain high order accuracy and non-oscillatory properties simultaneously. The main novelty of the new HWENO limiter in this paper is to reconstruct the polynomial on the target cell in a least square fashion [8] while the simple WENO limiter [29] is to use the entire polynomial of the original DG solutions in the neighboring cells with an addition of a constant for conservation. The modification in this paper improves the robustness in the computation of problems with strong shocks or contact discontinuities, without changing the compact stencil of the DG scheme. Numerical results for both one and two dimensional equations including Euler equations of compressible gas dynamics are provided to illustrate the viability of this modified limiter.     

A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations

A comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flows to DNS of turbulent flows, are presented to assess the performance of these schemes. Numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.     

A New Mapped WENO Scheme Using Order-Preserving Mapping

Existing mapped WENO schemes can hardly prevent spurious oscillations while preserving high resolutions at long output times. We reveal in this paper the essential reason of such phenomena. It is actually caused by that the mapping function in these schemes can not preserve the order of the nonlinear weights of the stencils. The nonlinear weights may be increased for non-smooth stencils and be decreased for smooth stencils. It is then indicated to require the set of mapping functions to be order-preserving in mapped WENO schemes. Therefore, we propose a new mapped WENO scheme with a set of mapping functions to be order-preserving which exhibits a remarkable advantage over the mapped WENO schemes in references. For long output time simulations of the one-dimensional linear advection equation, the new scheme has the capacity to attain high resolutions and avoid spurious oscillations near discontinuities meanwhile. In addition, for the two-dimensional Euler problems with strong shock waves, the new scheme can significantly reduce the numerical oscillations.     

On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes

In this paper we consider two commonly used classes of finite volume weighted essentially non-oscillatory (WENO) schemes in two dimensional Cartesian meshes. We compare them in terms of accuracy, performance for smooth and shocked solutions, and efficiency in CPU timing. For linear systems both schemes are high order accurate, however for nonlinear systems, analysis and numerical simulation results verify that one of them (Class A) is only second order accurate, while the other (Class B) is high order accurate. The WENO scheme in Class A is easier to implement and costs less than that in Class B. Numerical experiments indicate that the resolution for shocked problems is often comparable for schemes in both classes for the same building blocks and meshes, despite of the difference in their formal order of accuracy. The results in this paper may give some guidance in the application of high order finite volume schemes for simulating shocked flows.     

Arbitrarily High-Order (Weighted) Essentially Non-Oscillatory Finite Difference Schemes for Anelastic Flows on Staggered Meshes

We propose a WENO finite difference scheme to approximate anelastic flows, and scalars advected by them, on staggered grids. In contrast to existing WENO schemes on staggered grids, the proposed scheme is designed to be arbitrarily high-order accurate as it judiciously combines ENO interpolations of velocities with WENO reconstructions of spatial derivatives. A set of numerical experiments are presented to demonstrate the increase in accuracy and robustness with the proposed scheme, when compared to existing WENO schemes and state-of-the-art central finite difference schemes.     

Simulation of Two-Dimensional Scramjet Combustor Reacting Flow Field Using Reynolds Averaged Navier-Stokes WENO Solver

The non-equilibrium chemical reacting combustion flows of a proposed long slender scramjet system were numerically studied by solving the turbulent Reynolds averaged Navier-Stokes (RANS) equations. The Spalart-Allmaras one equation turbulence model is used which produces better results for near wall and boundary layer flow field problems. The lower-upper symmetric Gauss-Seidel implicit scheme, which enables results converge efficiently under steady state condition, is combined with the weighted essentially non-oscillatory (WENO) scheme to yield an accurate simulation tool for scramjet combustion flow field analysis. Using the WENO schemes high-order accuracy and its non-oscillatory solution at flow discontinuities, better resolution of the hypersonic flow problems involving complex shock-shock/shock-boundary layer interactions inside the flow path, can be achieved. Two types of scramjet combustor with cavity-based and strut-based fuel injector were considered as the testing models. The flow characteristics with and without combustion reactions of the two types of combustor models were studied with a transient hydrogen/oxygen combustion model. The detailed results of aerodynamic data are obtained and discussed, moreover, the combustion properties of varying the equivalent ratio of hydrogen, including the concentration of reacting species, hydrogen and oxygen, and the reacting products, water, are demonstrated to study the combustion process and performance of the combustor. The comparisons of flow field structures, pressure on wall and velocity profiles between the experimental data and the solutions of the present algorithms, showed qualitatively as well as the quantitatively in good agreement, and validated the adequacy of the present simulation tool for hypersonic scramjet reacting flow analysis.     

A Hybrid WENO Scheme for Steady Euler Equations in Curved Geometries on Cartesian Grids

For steady Euler equations in complex boundary domains, high-order shockcapturing schemes usually suffer not only from the difficulty of steady-state convergence but also from the problem of dealing with physical boundaries on Cartesian grids to achieve uniform high-order accuracy. In this paper, we utilize a fifth-order finite difference hybrid WENO scheme to simulate steady Euler equations, and the same fifth-order WENO extrapolation methods are developed to handle the curved boundary. The values of the ghost points outside the physical boundary can be obtained by applying WENO extrapolation near the boundary, involving normal derivatives acquired by the simplified inverse Lax-Wendroff procedure. Both equivalent expressions involving curvature and numerical differentiation are utilized to transform the tangential derivatives along the curved solid wall boundary. This hybrid WENO scheme is robust for steady-state convergence and maintains high-order accuracy in the smooth region even with the solid wall boundary condition. Besides, the essentially non-oscillation property is achieved. The numerical spectral analysis also shows that this hybrid WENO scheme has low dispersion and dissipation errors. Numerical examples are presented to validate the high-order accuracy and robust performance of the hybrid scheme for steady Euler equations in curved domains with Cartesian grids.     

Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation

In this paper, we propose a new conservative semi-Lagrangian (SL) finite difference (FD) WENO scheme for linear advection equations, which can serve as a base scheme for the Vlasov equation by Strang splitting [4]. The reconstruction procedure in the proposed SL FD scheme is the same as the one used in the SL finite volume (FV) WENO scheme [3]. However, instead of inputting cell averages and approximate the integral form of the equation in a FV scheme, we input point values and approximate the differential form of equation in a FD spirit, yet retaining very high order (fifth order in our experiment) spatial accuracy. The advantage of using point values, rather than cell averages, is to avoid the second order spatial error, due to the shearing in velocity (v) and electrical field (E) over a cell when performing the Strang splitting to the Vlasov equation. As a result, the proposed scheme has very high spatial accuracy, compared with second order spatial accuracy for Strang split SL FV scheme for solving the Vlasov-Poisson (VP) system. We perform numerical experiments on linear advection, rigid body rotation problem; and on the Landau damping and two-stream instabilities by solving the VP system. For comparison, we also apply (1) the conservative SL FD WENO scheme, proposed in [22] for incompressible advection problem, (2) the conservative SL FD WENO scheme proposed in [21] and (3) the non-conservative version of the SL FD WENO scheme in [3] to the same test problems. The performances of different schemes are compared by the error table, solution resolution of sharp interface, and by tracking the conservation of physical norms, energies and entropies, which should be physically preserved.     

Comparative Study of Three High Order Schemes for LES of Temporally Evolving Mixing Layers

Three high order shock-capturing schemes are compared for large eddy simulations (LES) of temporally evolving mixing layers for different convective Mach numbers ranging from the quasi-incompressible regime to highly compressible supersonic regime. The considered high order schemes are fifth-order WENO (WENO5), seventh-order WENO (WENO7) and the associated eighth-order central spatial base scheme with the dissipative portion of WENO7 as a nonlinear post-processing filter step (WENO7fi). This high order nonlinear filter method of Yee & Sjogreen is designed for accurate and efficient simulations of shock-free compressible turbulence, turbulence with shocklets and turbulence with strong shocks with minimum tuning of scheme parameters. The LES results by WENO7fi using the same scheme parameter agree well with experimental results compiled by Barone et al., and published direct numerical simulations (DNS) work of Rogers & Moser and Pantano & Sarkar, whereas results by WENO5 and WENO7 compare poorly with experimental data and DNS computations.     

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [11] for convection-diffusion equations, which relies on a special kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems. To overcome these difficulties, a new kernel-based formulation is designed to approach the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems, hence allowing much larger time step evolution compared with other explicit schemes. In addition, without extra computational cost, the proposed scheme can enlarge the available interval of the special parameter in the formulation, leading to less errors and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for one- and two-dimensional scalar and system, demonstrating the designed high order accuracy and unconditionally stable property of the scheme.     

Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning

Conservation laws are considered to be fundamental laws of nature. It has broad applications in many fields, including physics, chemistry, biology, geology, and engineering. Solving the differential equations associated with conservation laws is a major branch in computational mathematics. The recent success of machine learning, especially deep learning in areas such as computer vision and natural language processing, has attracted a lot of attention from the community of computational mathematics and inspired many intriguing works in combining machine learning with traditional methods. In this paper, we are the first to view numerical PDE solvers as an MDP and to use (deep) RL to learn new solvers. As proof of concept, we focus on 1-dimensional scalar conservation laws. We deploy the machinery of deep reinforcement learning to train a policy network that can decide on how the numerical solutions should be approximated in a sequential and spatial-temporal adaptive manner. We will show that the problem of solving conservation laws can be naturally viewed as a sequential decision-making process, and the numerical schemes learned in such a way can easily enforce long-term accuracy. Furthermore, the learned policy network is carefully designed to determine a good local discrete approximation based on the current state of the solution, which essentially makes the proposed method a meta-learning approach. In other words, the proposed method is capable of learning how to discretize for a given situation mimicking human experts. Finally, we will provide details on how the policy network is trained, how well it performs compared with some state-of-the-art numerical solvers such as WENO schemes, and supervised learning based approach L3D and PINN, and how well it generalizes.     

A Hermite WENO Method with Modified Ghost Fluid Method for Compressible Two-Medium Flow Problems

In this paper, we develop a novel approach by combining a new robust finite difference Hermite weighted essentially non-oscillatory (HWENO) method [51] with the modified ghost fluid method (MGFM) [25] to simulate the compressible two-medium flow problems. The main idea is that we first use the technique of the MGFM to transform a two-medium flow problem to two single-medium cases by defining the ghost fluids status based on the predicted interface status. Then the efficient and robust HWENO finite difference method is applied for solving the single-medium flow cases. By using immediate neighbor information to deal with both the solution and its derivatives, the fifth order finite difference HWENO scheme adopted in this paper is more compact and has higher resolution than the classical fifth order finite difference WENO scheme of Jiang and Shu [14]. Furthermore, by combining the HWENO scheme with the MGFM to simulate the two-medium flow problems, less ghost point information is needed than that in using the classical WENO scheme in order to obtain the same numerical accuracy. Various one-dimensional and two-dimensional two-medium flow problems are solved to illustrate the good performances of the proposed method.     

Piecewise Polynomial Mapping Method and Corresponding WENO Scheme with Improved Resolution

The method of mapping function was first proposed by Henrick et al. [J. Comput. Phys. 207:542-547 (2005)] to adjust nonlinear weights in [0,1] for the fifth-order WENO scheme, and through which the requirement of convergence order is satisfied and the performance of the scheme is improved. Different from Henrick's method, a concept of piecewise polynomial function is proposed in this study and corresponding WENO schemes are obtained. The advantage of the new method is that the function can have a gentle profile at the location of the linear weight (or the mapped nonlinear weight can be close to its linear counterpart), and therefore is favorable for the resolution enhancement. Besides, the function also has the flexibility of quick convergence to identity mapping near two endpoints of [0,1], which is favorable for improved numerical stability. The fourth-, fifth- and sixth-order polynomial functions are constructed correspondingly with different emphasis on aforementioned flatness and convergence. Among them, the fifth-order version has the flattest profile. To check the performance of the methods, the 1-D Shu-Osher problem, the 2-D Riemann problem and the double Mach reflection are tested with the comparison of WENO-M, WENO-Z and WENO-NS. The proposed new methods show the best resolution for describing shear-layer instability of the Riemann problem, and they also indicate high resolution in computations of double Mach reflection, where only these proposed schemes successfully resolved the vortex-pairing phenomenon. Other investigations have shown that the single polynomial mapping function has no advantage over the proposed piecewise one, and it is of no evident benefit to use the proposed method for the symmetric fifth-order WENO. Overall, the fifth-order piecewise polynomial and corresponding WENO scheme are suggested for resolution improvement.