We develop a conservative and positivity-preserving discontinuous Galerkin (DG) method for the population balance equation (PBE), which models the distribution of particle numbers across particle sizes due to growth, nucleation, aggregation, and breakage. To ensure number conservation in growth and mass conservation in aggregation and breakage, we design a DG scheme that applies standard treatment for growth and nucleation, and introduces a novel discretization for aggregation and breakage. The birth and death terms are discretized in a symmetric double-integral form, evaluated using a common refinement of the integration domain and carefully selected quadrature rules. Beyond conservation, we focus on preserving the positivity of the number density in aggregation–breakage. Since local mass corresponds to the first moment, the classical Zhang–Shu limiter, which preserves the zeroth moment (i.e., cell average), is not directly applicable. We address this by proving the positivity of the first moment on each cell and constructing a moment-conserving limiter that enforces nonnegativity across the domain. To the best of our knowledge, this is the first work to develop a positivity-preserving algorithm that conserves a prescribed moment. The numerical results verify the accuracy, conservation, and robustness of the proposed method.

Weighted essentially nonoscillatory (WENO) schemes are a popular class of numerical methods for solving hyperbolic conservation laws. Since WENO schemes are designed to deal with problems with both complicated solution structures and discontinuities / sharp gradient regions, their sophisticated nonlinear properties and high-order accuracy require more operations than many other schemes. The methodology of hybrid methods is an effective approach to decrease the computational costs and dissipation errors of WENO schemes and achieve better resolution. One of the key components for the success of hybrid WENO schemes is the application of a robust and efficient troubled-cell indicator, which detects the computational cells where the solution loses regularity. Recently, troubled-cell indicators based on artificial neural networks (ANNs) have been developed in the literature, which have the advantage of less dependence on tunable parameters and being more robust than many traditional troubled-cell indicators, and such ANN based troubled-cell indicators have been applied to hybrid finite difference WENO schemes effectively. Motivated by these works, in this paper we develop a hybrid finite volume WENO method with an ANN based troubled-cell indicator for solving hyperbolic conservation laws. While the finite difference WENO schemes are more efficient than the finite volume WENO schemes for multidimensional problems on uniform grids, the finite volume WENO schemes have the advantage such as being flexible and easy to apply on nonuniform grids. We introduce an ANN based troubled-cell indicator by constructing a multilayer perceptron (MLP) model, one of the most common ANN models. The third-order WENO scheme is focused in this paper. Extensive numerical experiments for solving various scalar equations with both convex and non-convex cases, and the Euler systems of equations on uniform and nonuniform grids of one-dimensional (1D) and two-dimensional (2D) domains, are performed to show the accuracy and nonlinear stability of the proposed hybrid finite volume WENO scheme with the MLP troubled-cell indicator. Significant accuracy improvement and computational-cost saving over the original WENO scheme are observed. Numerical experiments and comparisons with the widely-used KXRCF indicator also show the good performance of the MLP troubled-cell indicator. Although the MLP troubled-cell indicator is trained on uniform grids, it performs very well on nonuniform grids obtained by randomly perturbing uniform grids.

In this paper, we extend the previous work on absolutely convergent fixed-point fast sweeping WENO methods by Li et al. (J. Comput. Phys. 443: 110516, 2021) and design a fifth-order hybrid fast sweeping scheme for solving steady state problems of hyperbolic conservation laws. Unlike many other fast sweeping methods, the explicit property of fixed-point fast sweeping methods provides flexibility to apply the alternative weighted essentially non-oscillatory (AWENO) scheme with unequal-sized substencils as the local solver, which facilitates the usage of arbitrary monotone numerical fluxes. Furthermore, a novel hybrid technique is designed in the local solver to combine the nonlinear AWENO interpolation with the linear scheme for an additional improvement in efficiency of the high-order fast sweeping iterations. Numerical examples show that the developed fixed-point fast sweeping hybrid AWENO method with unequal-sized substencils can achieve absolute convergence (i.e., the residue of the fast sweeping iterations converges to machine zero / round off errors) more easily than the original AWENO method with equal-sized substencils, and is more efficient than the popular third-order total variation diminishing (TVD) Runge-Kutta time-marching approach to converge to steady state solutions.

In this paper, we focus on the finite difference approximation of nonlinear degenerate parabolic equations, a special class of parabolic equations where the viscous term vanishes in certain regions. This vanishing gives rise to additional challenges in capturing sharp fronts, beyond the restrictive CFL conditions commonly encountered with explicit time discretization in parabolic equations. To resolve the sharp front, we adopt the high-order multi-resolution alternative finite difference WENO (A-WENO) methods for the spatial discretization. To alleviate the time step restriction from the nonlinear stiff diffusion terms, we employ the exponential time differencing Runge-Kutta (ETD-RK) methods, a class of efficient and accurate exponential integrators, for the time discretization. However, for highly nonlinear spatial discretizations such as high-order WENO schemes, it is a challenging problem how to efficiently form the linear stiff part in applying the exponential integrators, since direct computation of a Jacobian matrix for high-order WENO discretizations of the nonlinear diffusion terms is very complicated and expensive. Here we propose a novel and effective approach of replacing the exact Jacobian of high-order multi-resolution A-WENO scheme with that of the corresponding high-order linear scheme in the ETD-RK time marching, based on the fact that in smooth regions the nonlinear weights closely approximate the optimal linear weights, while in non-smooth regions the stiff diffusion degenerates. The algorithm is described in detail, and numerous numerical experiments are conducted to demonstrate the effectiveness of such a treatment and the good performance of our method. The stiffness of the nonlinear parabolic partial differential equations (PDEs) is resolved well, and large time-step size computations are achieved.

In this paper, we study high-order exponential time differencing Runge-Kutta (ETD-RK) discontinuous Galerkin (DG) methods for nonlinear degenerate parabolic equations. This class of equations exhibits hyperbolic behavior in degenerate regions and parabolic behavior in non-degenerate regions, resulting in sharp wave fronts in the solution profiles and a parabolic-type time-step restriction, τ∼ O(h^2), for explicit time integration. To address these challenges and solve such equations in complex domains, we employ DG methods with appropriate stabilizing limiters on unstructured meshes to capture the wave fronts and use ETD-RK methods for time integration to resolve the stiffness of parabolic terms. We extract the system's stiffness using the Jacobian matrix of the DG discretization for diffusion terms and adopt a nodal formulation to facilitate its computation. The algorithm is described in detail for two-dimensional triangular meshes. We also conduct a linear stability analysis in one spatial dimension and present computational results on three-dimensional simplex meshes, demonstrating significant improvements in stability and large time-step sizes.

In this paper, we investigate the stability and time-step constraints for solving advection-diffusion equations using exponential time differencing (ETD) Runge-Kutta (RK) methods in time and discontinuous Galerkin (DG) methods in space. We demonstrate that the resulting fully discrete scheme is stable when the time-step size is upper bounded by a constant. More specifically, when central fluxes are used for the advection term, the schemes are stable under the time-step constraint tau <= tau_0 * d / a^2, while when upwind fluxes are used, the schemes are stable if tau <= maxtau_0 * d / a^2, c_0 * h / a. Here, tau is the time-step size, h is the spatial mesh size, and a and d are constants for the advection and diffusion coefficients, respectively. The constant c_0 is the CFL constant for the explicit RK method for the purely advection equation, and tau_0 is a constant that depends on the order of the ETD-RK method. These stability conditions are consistent with those of the implicit-explicit RKDG method. The time-step constraints are rigorously proved for the lowest-order case and are validated through Fourier analysis for higher-order cases. Notably, the constant tau_0 in the fully discrete ETD-RKDG schemes appears to be determined by the stability condition of their semi-discrete (continuous in space, discrete in time) ETD-RK counterparts and is insensitive to the polynomial degree and the specific choice of the DG method. Numerical examples, including problems with nonlinear convection in one and two dimensions, are provided to validate our findings.

Fast sweeping methods are a class of efficient iterative methods developed in the literature to solve steady-state solutions of hyperbolic partial differential equations (PDEs). In Zhang et al. (J Sci Comput 29:25–56, 2006) and Xiong et al. (J Sci Comput 45:514–536, 2010), high order accuracy fast sweeping schemes based on classical weighted essentially non-oscillatory (WENO) local solvers were developed for solving static Hamilton–Jacobi equations. However, since high order classical WENO methods (e.g., fifth order and above) often suffer from difficulties in their convergence to steady-state solutions, iteration residues of high order fast sweeping schemes with these local solvers may hang at a level far above round-off errors even after many iterations. This issue makes it difficult to determine the convergence criterion for the high order fast sweeping methods and challenging to apply the methods to complex problems. Motivated by the recent work on absolutely convergent fast sweeping method for steady-state solutions of hyperbolic conservation laws in Li et al. (J Comput Phys 443:110516, 2021), in this paper we develop high order fast sweeping methods with multi-resolution WENO local solvers for solving Eikonal equations, an important class of static Hamilton–Jacobi equations. Based on such kind of multi-resolution WENO local solvers with unequal-sized sub-stencils, iteration residues of the designed high order fast sweeping methods can settle down to round-off errors and achieve the absolute convergence. In order to obtain high order accuracy for problems with singular source-point, we apply the factored Eikonal approach developed in the literature and solve the resulting factored Eikonal equations by the new high order WENO fast sweeping methods. Extensive numerical experiments are performed to show the accuracy, computational efficiency, and advantages of the new high order fast sweeping schemes for solving static Hamilton–Jacobi equations.

High order accuracy fast sweeping methods have been developed in the literature to efficiently solve steady state solutions of hyperbolic partial differential equations (PDEs) on rectangular meshes, while they were not available yet on unstructured meshes. In this paper, we extend high order accuracy fast sweeping methods to unstructured triangular meshes by applying fixed-point iterative sweeping techniques to a fifth-order finite volume unstructured WENO scheme, for solving steady state solutions of hyperbolic conservation laws. Similar as other fast sweeping methods, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady state solutions. An advantage of fixed-point fast sweeping methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. This also provides certain convenience in designing high order fast sweeping methods on unstructured meshes. As in the first order fast sweeping methods on triangular meshes, we introduce multiple reference points to determine alternating sweeping directions on unstructured meshes. All the cells on the mesh are ordered according to their centroids' distances to those reference points, and the resulted orderings provide sweeping directions for iterations. To make the residue of the fast sweeping iterations converge to machine zero / round off errors, we follow the approach in our early work of developing the absolutely convergent fixed-point fast sweeping WENO methods on rectangular meshes, and adopt high order WENO scheme with unequal-sized sub-stencils, specifically here a fifth-order finite volume unstructured WENO scheme for spatial discretization. Extensive numerical experiments, including problems with complex domain geometries, are performed to show the accuracy, computational efficiency, and absolute convergence of the presented fifth-order fast sweeping scheme on triangular meshes. Furthermore, the proposed method is compared with the forward Euler time marching method and the popular third order total variation diminishing Rung-Kutta (TVD-RK3) time-marching method for steady state computations. Numerical examples show that the developed fixed-point fast sweeping WENO method is the most efficient scheme among them, and especially it can save up to 70% CPU time costs than TVD-RK3 in order to converge to steady state solutions.

Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations (PDEs). The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions. A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. Hence, they are easy to be applied to a general hyperbolic system. To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains, inverse Lax-Wendroff (ILW) procedures were developed as a very effective approach in the literature. In this paper, we combine a fifth-order fixed-point fast sweeping WENO method with an ILW procedure to solve steady-state solution of hyperbolic conservation laws on complex computing regions. Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids. Numerical results show high-order accuracy and good performance of the method. Furthermore, the method is compared with the popular third-order total variation diminishing Runge-Kutta (TVD-RK3) time-marching method for steady-state computations. Numerical examples show that for most of examples, the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.

High-order accurate weighted essentially non-oscillatory (WENO) schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations (PDEs). Due to highly nonlinear property of the WENO algorithm, large amount of computational costs are required for solving multidimensional problems. In our previous work (Lu et al. in Pure Appl Math Q 14: 57–86, 2018; Zhu and Zhang in J Sci Comput 87: 44, 2021), sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations, and it was shown that significant CPU times were saved, while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids. In this technical note, we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme, which has very interesting properties such as its simplicity in linear weights’ construction over a classical WENO scheme. Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times, and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.

We develop a second-order continuous finite element method for solving the static Eikonal equation. It is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear system. More specifically, the homotopy method is utilized to decrease the viscosity coefficient gradually, while Newton???s method is applied to compute the solution for each viscosity coefficient. Newton???s method alone converges for just big enough viscosity coefficients on very coarse grids and for simple 1D examples, but the proposed method is much more robust and guarantees the convergence of the nonlinear solver for all viscosity coefficients and for all examples over all grids. Numerical experiments from 1D to 3D are presented to confirm the second-order convergence and the effectiveness of the proposed method on both structured or unstructured meshes.

In [F. Li, C.-W. Shu, Y.-T. Zhang, H. Zhao, Journal of Computational Physics 227 (2008) 8191-8208], we developed a fast sweeping method based on a hybrid local solver which is a combination of a discontinuous Galerkin (DG) finite element solver and a first order finite difference solver for Eikonal equations. The method has second order accuracy in the L 1 norm and a very fast convergence speed, but only first order accuracy in the L ∞ norm for the general cases. This is an obstacle to the design of higher order DG fast sweeping methods. In this paper, we overcome this problem by developing uniformly accurate DG fast sweeping methods for solving Eikonal equations. We design novel causality indicators which guide the information flow directions for the DG local solver. The values of these indicators are initially provided by the first order finite difference fast sweeping method, and they are updated during iterations along with the solution. We observe both a uniform second order accuracy in the L ∞ norm (in smooth regions) and the fast convergence speed (linear computational complexity) in the numerical examples.

Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations (PDEs). As other types of fast sweeping schemes, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order. The resulting iterative schemes have a fast convergence rate to steady-state solutions. Moreover, an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system. Hence, they are robust and flexible, and have been combined with high-order accurate weighted essentially non-oscillatory (WENO) schemes to solve various hyperbolic PDEs in the literature. For multidimensional nonlinear problems, high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs. In this technical note, we apply sparse-grid techniques, an effective approximation tool for multidimensional problems, to fixed-point fast sweeping WENO methods for reducing their computational costs. Here, we focus on fixed-point fast sweeping WENO schemes with third-order accuracy (Zhang et al. 2006 [41]), for solving Eikonal equations, an important class of static Hamilton-Jacobi (H-J) equations. Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes, and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.

In this paper, we develop a simplified hybrid weighted essentially non-oscillatory (WENO) method combined with the modified ghost fluid method (MGFM) [28] to simulate the compressible two-medium flow problems. The MGFM can turn the two-medium flow problems into two single-medium cases by defining the ghost fluids status in terms of the predicted the interface status, which makes the material interface "invisible". For the single medium flow case, we adapt between the linear upwind scheme and the WENO scheme automatically by identifying the regions of the extreme points for the reconstruction polynomial as same as the hybrid WENO scheme [50]. Instead of calculating their exact locations, we only need to know the regions of the extreme points based on the zero point existence theorem, which is simpler for implementation and saves computation time. Meanwhile, it still keeps the robustness and has high efficiency. Extensive numerical results for both one and two dimensional two-medium flow problems are performed to demonstrate the good performances of the proposed method.

Fixed-point iterative sweeping methods were developed in the literature to efficiently solve steady state solutions of Hamilton-Jacobi equations and hyperbolic conservation laws. Similar as other fast sweeping schemes, the key components of this class of methods are the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. Furthermore, good properties of fixed-point iterative sweeping methods include that they have explicit forms and do not involve inverse operation of nonlinear local systems, and they can be applied to general hyperbolic equations using any monotone numerical fluxes and high order approximations easily. In [L. Wu, Y.-T. Zhang, S. Zhang and C.-W. Shu, Commun. Comput. Phys., 20 (2016)], a fifth order fixed-point sweeping WENO scheme was designed and it was shown that the scheme converges much faster than the total variation diminishing (TVD) Runge-Kutta approach by stability improvement of high order schemes with a forward Euler time-marching. An open problem is that for some benchmark numerical examples, the iteration residue of the fixed-point sweeping WENO scheme hangs at a truncation error level instead of settling down to machine zero. This issue makes it difficult to determine the convergence criterion for the iteration and challenging to apply the method to complex problems. To solve this issue, in this paper we apply the multi-resolution WENO scheme developed in [J. Zhu and C.-W. Shu, J. Comput. Phys., 375 (2018)] to the fifth order fixed-point sweeping WENO scheme and obtain an absolutely convergent fixed-point fast sweeping method for steady state of hyperbolic conservation laws, i.e., the residue of the fast sweeping iterations converges to machine zero / round off errors for all benchmark problems tested.

The weighted essentially non-oscillatory (WENO) schemes, especially the fifth order WENO schemes, are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important question. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In a recent work [Lu, Chen and Zhang, Pure and Applied Mathematics Quarterly, 14 (2018) 57-86], a third order finite difference WENO method with sparse-grid combination technique was designed to solve multidimensional hyperbolic equations including both linear advection equations and nonlinear Burgers’ equations. Numerical experiments showed that WENO computations on sparse grids achieved comparable third order accuracy in smooth regions of the solutions and nonlinear stability as that for computations on regular single grids. In application problems, higher than third order WENO schemes are often preferred in order to efficiently resolve the complex solution structures. In this paper, we extend the approach to higher order WENO simulations specifically the fifth order WENO scheme. A fifth order WENO interpolation is applied in the prolongation part of the sparse-grid combination technique to deal with discontinuous solutions. Benchmark problems are first solved to show that significant CPU times are saved while both fifth order accuracy and stability of the WENO scheme are preserved for simulations on sparse grids. The fifth order sparse grid WENO method is then applied to kinetic problems modeled by high dimensional Vlasov based PDEs to further demonstrate large savings of computational costs by comparing with simulations on regular single grids. Several open problems are discussed at last.

In this paper, we develop a modified fifth order accuracy finite difference Hermite WENO (HWENO) scheme for solving hyperbolic conservation laws. The main idea is that we first modify the derivatives of the solution by Hermite WENO interpolations, then we discretize the original and derivative equations in the spatial directions by the same approximation polynomials. Comparing with the original finite difference HWENO scheme of Liu and Qiu (J Sci Comput 63:548–572, 2015), one of the advantages is that the modified HWENO scheme is more robust than the original one since we do not need to use the additional positivity-preserving flux limiter methodology, and larger CFL number can be applied. Another advantage is that higher order numerical accuracy than the original scheme can be achieved for two-dimensional problems under the condition of using the same approximation stencil and information. Furthermore, the modified scheme preserves the nice property of compactness shared by HWENO schemes, i.e., only immediate neighbor information is needed in the reconstruction, and it has smaller numerical errors and higher resolution than the classical fifth order finite difference WENO scheme of Jiang and Shu (J Comput Phys 126:202–228, 1996). Various benchmark numerical tests of both one-dimensional and two-dimensional problems are presented to illustrate the numerical accuracy, high resolution and robustness of the proposed novel HWENO scheme.

This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schrödinger (NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grünwald-Letnikov difference (WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schrödinger (FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations (ODEs) in matrices formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson (CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor (cIIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O (τ 2 ). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.

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 weighted essentially non-oscillatory (WENO) schemes are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). The computational cost of such schemes increases significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points and nonlinearity of high order accuracy WENO schemes. How to achieve fast computations by WENO methods for high spatial dimension PDEs is a challenging and important question. Recently, sparse-grid has become a major approximation tool for high dimensional problems. The open question is how to design WENO computations on sparse grids such that comparable high order accuracy of WENO schemes in smooth regions and essentially non-oscillatory stability in non-smooth regions of the solutions can still be achieved as that for computations on regular single grids? In this paper, we combine the third order finite difference WENO method with sparse-grid combination technique and solve high spatial dimension hyperbolic equations on sparse grids. WENO interpolation is proposed for the prolongation part in sparse grid combination techniques to deal with discontinuous solutions of hyperbolic equations. Numerical examples are presented to show that significant computational times are saved while both high order accuracy and stability of the WENO scheme are maintained for simulations on sparse grids.