A New Class of HighOrder Methods for Fluid Dynamics Simulations Using Gaussian Process Modeling: OneDimensional Case
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Abstract
We introduce an entirely new class of highorder methods for computational fluid dynamics based on the Gaussian process (GP) family of stochastic functions. Our approach is to use kernelbased GP prediction methods to interpolate/reconstruct highorder approximations for solving hyperbolic PDEs. We present a new highorder formulation to solve (magneto)hydrodynamic equations using the GP approach that furnishes an alternative to conventional polynomialbased approaches.
Keywords
Gaussian processes Stochastic models Highorder methods Finite volume method Gas dynamics Magnetohydrodynamics1 Introduction
Cutting edge simulations of gas dynamics and magnetohydrodynamics (MHD) have been among the headliner applications of scientific highperformance computing (HPC) [12, 13, 32, 61]. They are expected to remain important as new HPC architectures with ever more powerful capabilities come online in the decades to come. A notable trend in recent HPC developments concerns the hardware design of the newer architectures. It is expected that newer HPC architectures will feature a radical change in the balance between computation and memory resources, with memory per compute core declining dramatically from current levels [2, 12, 61]. This trend tells us that new algorithmic strategies will be required to meet the goals of saving memory and accommodating increased computation. This paradigm shift in designing scientific algorithms has become a great importance in HPC applications. In the context of numerical methods for computational fluid dynamics (CFD), one desirable approach is to design highorder accurate methods [61] that, in contrast to loworder methods, can achieve an increased target solution accuracy more efficiently and more quickly by computing increased higherorder floatingpoint approximations on a given grid resolution [25, 36, 37]. This approach embodies in a concrete manner the desired tradeoff between memory and computation by exercising more computation per memory—or equivalently, the equal amount of computation with less memory.
Within the broad framework of finite difference method (FDM) and finite volume method (FVM) discretizations, discrete algorithms of data interpolation and reconstruction play a key role in numerical methods for PDE integration [36, 37, 62]. They are frequently the limiting factor in the convergence rate, efficiency, and algorithmic complexity of a numerical scheme in FDM [26, 43] and in FVM [7, 35, 36, 42, 55, 62]. More generally, interpolation and reconstruction are not only essential for estimating highorder accurate approximations for fluxes at quadrature points on each cell but also for interface tracking; for prolonging states from coarse zones to corresponding refined zones in adaptivemesh refinement (AMR) schemes; and for various other contexts associated with highorder solutions. In CFD simulations, these interpolation and reconstruction algorithms must be carried out as accurately as possible, because, to a large extent, their accuracy is one of the key factors that determine the overall accuracy of the simulation.
Polynomialbased approaches are the most successful and popular among interpolation/reconstruction methods in this field. There are a couple of convincing reasons for this state of affairs. First, they are easily relatable to Taylor expansion, the most familiar of function approximations. Second, the nominal Nth order accuracy of polynomial interpolation/reconstruction is derived from using polynomials of degree \((N1)\), bearing a leading term of the error that scales with \({\mathcal {O}}(\varDelta ^{N})\) as the local grid spacing \(\varDelta \) approaches to zero [36, 37, 62]. However, the simplicity of polynomial interpolation/reconstruction comes at a price: the polynomial approach is notoriously prone to oscillations in data fitting, especially with discontinuous data [20]; furthermore, in many practical situations the highorder polynomial interpolation/reconstruction must be carried out on a fixed size of stencils, whereby there is a onetoone relationship between the order of the interpolation/reconstruction and the size of the stencils. This becomes a restriction in particular when unstructured meshes are considered in multiple spatial dimensions. Lastly, another related major issue lies in the fact that the algorithmic complexity of such polynomial based schemes typically grows with an order of accuracy [18], as well as with a spatial dimensionality [7, 42, 73] in FVM.
To overcome the aforementioned issues in polynomial methods, practitioners have developed the socalled “nonpolynomial” interpolation/reconstruction based on the meshfree Radial Basis Function (RBF) approximations in the last decades. The core idea is to replace the polynomial interpolants with RBFs, which is a part of a very general class of approximants from the field known as Optimal Recovery (OR) [68]. Several interpolation techniques in OR have shown practicable in the framework of solving hyperbolic PDEs [31, 46, 58], parabolic PDEs [44, 45], diffusion and reactiondiffusion PDEs [53], boundary value problems of elliptic PDEs [40], interpolations on irregular domains [9, 24, 41], and also interpolations on a more general set of scattered data [17]. Historically, RBFs were introduced to seek exact function interpolations [47]. Recently, the RBF approximations have been combined with the key ideas of handling discontinuities in the ENO [23] and WENO [26, 39] methods. Such approaches, termed as ENO/WENORBF, have been extended to solve nonlinear scalar equations and the Euler equations in the FDM [29] and the FVM [21] frameworks. These studies focused on designing their RBF methods with the use of adaptive shape parameters to control local errors. Also in [3], two types of multiquadrics and polyharmonic spline RBFs were used to model the Euler equations with a strategy of selecting optimal shape parameters for different RBF orders. Stability analysis on the fully discretized hyperbolic PDEs in both space and time using the multiquadrics RBF is reported in [8]. Admittedly, while there exist a few conceptual resemblances between these RBF approaches and our new GP method, there are fundamental differences that distinguish the two approaches, which will be discussed later in this paper.
The goal of this article is to develop a new class of highorder methods that overcome the aforementioned difficulties in polynomial approaches by exploiting the alternative perspective afforded by GP modeling, a methodology borrowed from the field of statistical data modeling. In view of the novelty of our approach, which bridges the two distinct research fields of statistical modeling and CFD, it is our intention in this paper to first construct a mathematical formulation in a 1D framework. The current study will serve as a theoretical foundation for multidimensional extensions of the GP modeling for CFD applications, the topics of which will be studied in our future work.
 1.
GP interpolation that works on pointwise values of \(q(x_i)\) as both inputs and outputs, and
 2.
GP reconstruction that works on volumeaveraged values \(\langle q_i\rangle =\frac{1}{\varDelta {x}}\int _{\varDelta {x}}q(x,t^n)dx\) as inputs, reconstructing pointwise values as outputs.
2 Gaussian Process Modeling
The theory of GP, and more generally of stochastic functions, dates back to the work of Wiener [69] and Kolmogorov [33]. Modernday applications are numerous. Just in the physical sciences, GP prediction is in common use in meteorology, geology, and timeseries analysis [4, 48, 67], and in cosmology, where GP models furnish the standard description of the Cosmic Microwave Background [5]. Applications abound in many other fields, in particular, wherever spatial or timeseries data requires “nonparametric” modeling [4, 48, 59]. From the perspective of CFD applications, our specific goal in this study is to use the predictive GP modeling that is processed by training observed data, (e.g., cellaveraged fluid variables at cell centers) to produce a “datainformed” prediction (e.g., pointwise Riemann state values at cell interfaces). In what follows we give a brief overview on GP from the statistical perspective (Sect. 2.1), followed by our strategies of tuning GP for highorder interpolation (Sect. 2.2) and reconstruction (Sect. 2.3) in CFD applications. Readers who wish to pursue the subject in greater detail are referred to “Appendix A” toward the end of this paper as well as [4, 48, 59]. Readers who are more interested in our new GP interpolation and reconstruction algorithms in the framework of FDM and FVM than in the basic Bayesian theory may start reading our paper from Sect. 2.2 after briefly reviewing Eqs. (1) and (3) in Sect. 2.1.
2.1 GP—Statistical Perspective
GP is a class of stochastic processes, i.e., processes that sample functions (rather than points) from an infinite dimensional function space. Formally, a GP is a collection of random variables, any finite collection of which has a joint Gaussian distribution [4, 48]. For the purposes of interpolation/reconstruction, this would be the collection of input data, either pointwise values \(q(x_i)\) or the volume averaged \(\langle q_i \rangle \). In the language of GP, the input data form a joint Gaussian distribution with function values that are to be interpolated/reconstructed. In this way, the known input data (e.g., \(q(x_i)\) or \(\langle q_i \rangle \)) can be used to “train” a probability distribution for the unknown function values (e.g., \(q(x_*\)) at any arbitrary point \(x_*\)), with a posterior mean and uncertainty that are compatible with the observed data. The updated mean value is our target interpolation/reconstruction for FDM and FVM.

a mean function \(\bar{f}({\mathbf {x}}) = {\mathbb {E}}[f({\mathbf {x}})]\) over \({\mathbb {R}}^N\), and

a covariance kernel function which is a symmetric, positivedefinite integral kernel \(K({\mathbf {x}},\mathbf {y}) = {\mathbb {E}}[ \left( f({\mathbf {x}})\bar{f}({\mathbf {x}})\right) \left( f(\mathbf {y})\bar{f}(\mathbf {y})\right) ]\) over \({\mathbb {R}}^{N}\times {\mathbb {R}}^{N}\).
Given the function samples \({\mathbf {f}}=[f({\mathbf {x}}_{1}),\ldots ,f({\mathbf {x}}_{N})]^{T}\) obtained at spatial points \({\mathbf {x}}_{i}\), \(i=1,\ldots , N\), GP predictions aim to make a probabilistic statement about the value \(f_{*}\equiv f({\mathbf {x}}_{*})\) of the unknown function \(f\sim \mathcal {GP}(\bar{f},K)\) at a new spatial point \({\mathbf {x}}_{*}\). This is particularly of interest to us from the perspectives of FDM and FVM, because we can use GP to predict an unknown function value at cell interfaces (e.g., \(q_{i\pm \frac{1}{2}}\) in 1D) where both FDM and FVM require estimates of flux functions.
2.2 HighOrder GP Interpolation for CFD
In this paper, we are most interested in developing a highorder reconstruction method for FVM in which the function samples \({\mathbf {f}}\) are given as volumeaveraged data \(\langle q_i \rangle \). However, we first consider an interpolation method which uses pointwise data \(q_i\) as function samples \({\mathbf {f}}\). An algorithmic design of GP interpolation using pointwise data will provide a good mathematical foundation for FVM which reconstructs pointwise values from volumeaveraged data.
The mean \({\tilde{f}_{*}}\) of the distribution given in Eq. (3) is our interpolation of the function f at the point \({\mathbf {x}}_{*} \in {\mathbb {R}}^D\), \(D=1, 2, 3\), where f is any given fluid variable such as density, pressure, velocity fields, magnetic fields, etc. For the purpose of exposition, let us use q to denote one of such fluid variables (e.g., density \(\rho = \rho ({\mathbf {x}},t^n)\)). Also, the mathematical descriptions of both interpolation and reconstruction will be considered in 1D hereafter.
Since the matrix \({\mathbf {K}}\) is symmetric positivedefinite, the inversion of \({\mathbf {K}}\) can be obtained efficiently by Cholesky decomposition which is about a factor 2 faster than the usual LU decomposition. The decomposition is needed only once, at initialization time, for the calculation of the vector of weights \({\mathbf {w}}\).
An important feature of GP interpolation is that it naturally supports multidimensional stencil configurations. The reason for this is that there are many valid covariance functions over \(\mathbf {R}^D\) that are isotropic, and therefore do not bias interpolations and reconstructions along any special direction. The possibility of directionallyunbiased reconstruction over multidimensional stencils is a qualitative advantage of GP interpolation, especially when designing highorder algorithms [7, 42, 73]. Such highorder multidimensional GP methods will be studied in our future work.
2.3 HighOrder GP Reconstruction for CFD
The GP interpolation method outlined in Sect. 2.2 should, therefore, be modified so that reconstruction may account for such data type changes in both FDM and FVM. Note that the integral averages over a grid cell constitute “linear” operations on a function \(f({\mathbf {x}})\). Similar to the case with ordinary finitedimensional multivariate Gaussian distributions where linear operations on Gaussian random variables result in new Gaussian random variables with linearly transformed means and covariances, a set of N linear functionals operating on a GPdistributed function f has an Ndimensional Gaussian distribution with mean and covariance that are linear functionals of the GP mean function and the covariance function.
2.4 The GPSE Model
The latter two, \(\varSigma ^{2}\) and \(\ell \), are called “hyperparameters” which are the parameters that are built into the kernel. The hyperparameter \(\varSigma ^{2}\) has no effect on the posterior mean function, so one can set \(\varSigma ^{2}=1\) for simplicity. On the other hand, the hyperparameter \(\ell \) is the correlation length scale of the model. It determines the length scale of variation preferred by the GP model. Our GP predictions for interpolation/reconstruction, which necessarily agrees with the observed values of the function at the training points \(\mathbf {x}_k\), may “wiggle” on this scale between training points. In this sense, \(\ell \) is a “rigidity”, controlling the curvature scales of the prediction, that should correspond to the physical length scales of the features GP is to resolve. Since we want function interpolations/reconstructions that are smooth on the scale of the grid, we certainly want \(\ell >\varDelta \), and would also prefer \(\ell \ge R\). As just mentioned, the choice of \(\ell \) requires a balance between the physical length scales in the problem and the grid scale. This implies that different values of \(\ell \) can be employed depending on differing length scales on different regions of the computational domain. For this paper, we select a constant value for \(\ell \), which has a direct impact on the solution accuracy, as shown in Fig. 3c. The accuracy of the scheme is intimately tied to the scales of the flow and ideally, would need to be recalculated throughout a simulation to best capture the features specific to the problem at hand. Such a scheme would require additional mathematical and algorithmic considerations and we leave investigation on best practice for determining such an optimal \(\ell \) within the context of FDM/FVM reconstructions to our forthcoming papers. We have found that values of \(\ell /\varDelta =6\) or 12 to perform robustly and accurately under the grid and stencil sizes considered in this paper.
The relationship often leads to the tradeoff principle in that there is a conflicting balance between the obtainable accuracy and the numerical stability due to a large condition number of the kernel matrix when a grid is highly refined. For instance, in our GPSE model, SE suffers a notorious singularity issue where the kernel is prone to yield nearly singular matrices when the distance between any two points \({\mathbf {x}}\) and \(\mathbf {y}\) becomes progressively smaller (or equivalently, the grid becomes progressively refined in CFD).
In the statistical modeling community, a practical and wellknown fix for this problem is to add a “nugget” [11], i.e., a small perturbation \(c_o \mathbf {I}\) (where \(c_0\) is a small positive constant, usually chosen to be 10100 times the floatingpoint machine precision) is added to \({\mathbf {K}}\), where \(\mathbf {I}\) is the identity matrix. Unfortunately, this trick does not resolve the issue in the desired way because it often results in less accurate data predictions in GP. In the RBF community, the tradeoff principle also occurs in terms of the shape parameters of the RBFs and it has been investigated by several authors [15, 17, 22, 49]. More recent studies [3, 21, 29] have focused on finding their optimal values of the shape parameters in the combined ENO/WENORBF framework. On the other hand, in a series of studies by Wright and Fornberg [16, 71], the authors proposed an alternative way called ContourPadé algorithm of evaluating RBF approximation methods. The approach circumvents the direct evaluation of the RBF functions (RBFDirect) which suffers from severely illconditioned linear systems due to large condition numbers when the associated RBF functions become flat. Recently, a new improved ContourPadé algorithm has been proposed by using rational approximations of vectorvalued analytic functions [72]. Certainly, the ContourPadé approach can be employed in modeling our GP approach to improve the analogous situations of illconditioned GP covariance function matrices in Eqs. (3) and (17). A careful investigation of which will be discussed in our forthcoming papers. Our current strategy of this issue is given in Sect. 5.2.
The SE covariance function has two desirable properties [4, 11, 48]. First, it has the property of having a native space of \(C^{\infty }\) functions, which implies that the resulting interpolants converge exponentially with stencil size, for data sampled from smooth functions. Second, the SE kernel facilitates dimensional factorization, which is useful in multidimensional cases.
On the negative side, SE performs poorly when the dataset contains discontinuities, such as shocks and contact discontinuities in a CFD context. An intuitive resolution to this issue, from a statistical modeling perspective, would be to use other types of nonsmooth kernel functions instead such as the Matérn class, exponentialtypes, rational quadratic functions, Wendland, etc. [4, 48, 59]. They are known to be bettersuited for discontinuous datasets than SE by relaxing the strong smoothness assumptions of SE. By construction, the latent (or prior) sample function spaces generated by these nonsmooth kernels possess finiteorder of differentiability. As a result, we have seen that the posterior GP predictions of interpolations and reconstructions using such kernels turn out to be very noisy and rough, and their solutions often suffer from reduced solutionaccuracy and numerical instability. A covariance kernel function such as the Gibbs covariance or the neuralnetwork covariance [48] would be more appropriate for resolving discontinuities. The possibility of using these GP kernels for highorder interpolations and reconstructions is under investigation and will be studied in our future work. To deal with discontinuous solutions in this paper, we introduce yet another new approach to design an improved GP reconstruction algorithm, termed as “GPWENO”, which is described in Sect. 2.5. This new approach formulates a new set of nonpolynomial, GPbased smoothness indicators for discontinuous flows.
2.5 GPWENO: New GPBased Smoothness Indicators for NonSmooth Flows
For smooth flows the GP linear prediction in Eq. (17) with the SE covariance kernel Eq. (21) furnishes a highorder GP reconstruction algorithm without any extra controls on numerical stability. However, for nonsmooth flows, the unmodified GPSE reconstruction suffers from unphysical oscillations that originate at discontinuities such as shocks. To handle such issues with nonsmooth flows, we adopt the principal idea of employing the nonlinear weights in the Weighted Essentially Nonoscillatory (WENO) methods [26], by which we adaptively change the size of the reconstruction stencil to avoid interpolating through a discontinuity, while retaining highorder properties in smooth flow regions. A traditional WENO takes the weighted combination of candidate stencils based on the local smoothness of the individual substencils. The weights are chosen so that they are optimal in smooth regions in the sense that they are equivalent to an approximation using the global stencil that is the union of the candidate substencils.
It was observed by Jiang and Shu [26] that it is critically important to have appropriate smoothness indicators so that in smooth regions the nonlinear weights reproduce the optimal linear weights to the design accuracy in order to maintain the highorder of accuracy in smooth solutions. Typically this is shown by Taylor expansion of smoothness indicators in Eq. (35). Here we propose to calculate the eigendecomposition in Eq. (65) numerically, and therefore we don’t provide a mathematical proof of the accuracy of our new smoothness indicators. Nonetheless, we observe the expected high order convergence expected from the polynomial WENO reconstructions on the equivalent stencils in numerical experiments provided in Sect. 5.2.
3 StepbyStep Procedures of the GPSE Algorithm for 1D FVM
 1.PreSimulation: The following steps are carried out before starting a simulation, and any calculations therein need only be performed once, stored, and used throughout the actual simulation.At this point, before beginning the simulation, if quadruple precision was used in Step (b) and Step (c), the GP weights and linear weights can be truncated to double precision for use in the actual reconstruction step.
 (a)
Configure computational grid: Determine a GP stencil radius R as well as choose the size of the hyperparameter \(\ell \). This determines the SE kernel function in Eq. (21) as well as the global and candidate stencils in Eqs. (24) and (25).
 (b)
Compute GP weights: Compute the covariance matrices, \(\mathbf {C}\) and \(\mathbf {C}^m\) (Eq. 15) on the stencils \(S_R\) and each of \(S_m\), respectively. Compute the prediction vectors, \(\mathbf {T}_*\) and \(\mathbf {T}_*^m\) respectively on \(S_R\) and each of \(S_m\) (Eq. 16). The GP weight vectors, \(\mathbf {z}^T=\mathbf {T}_*\mathbf {C}^{1}\) on \(S_R\) and \(\mathbf {z}^T_m=\mathbf {T}_*^m(\mathbf {C}^m)^{1}\) on each of \(S_m\), can then be stored for use in the GP reconstruction. The columns of the matrices \(\mathbf {Z}_m\), \(\mathbf {z}_m(x_l)\), should be computed here for use in step (d).
It is crucial in this step as well as in Step (c) to use the appropriate floating point precision to prevent the covariance matrices, \(\mathbf {C}\) and \(\mathbf {C}^m\), from being numerically singular. This is discussed in more detail in Sect. 5.2. We find that double precision is only suitable up to condition numbers \(\kappa \sim 10^8\), whereas quadruple precision comfortably allows up to \(\kappa \sim 10^{18}\) (hence our choice in this paper). The standard double precision is used except for Step (b) and Step (c).
 (c)
Compute linear weights: Use the GP weight vectors to calculate and store the optimal linear weights \(\gamma _m\) according to Eq. (31).
 (d)
Compute kernel eigensystem: The eigensystem for the covariance matrices used in GPWENO are calculated using Eq. (65). The matrices, \(\mathbf {C}^m\), are the same on each of the candidate stencils in the GPWENO scheme presented here, so only one eigensystem needs to be determined. This eigensystem is then used to calculate and store the vectors \(\mathbf {P}^{(m)}_i\) in Eq. (43) for use in determining the smoothness indicators in the reconstruction Step 2.
 (a)
 2.
Reconstruction: Start a simulation. Choose \(f_0\) according to Eq. (62) or simply set to zero. The simplest choice \(f_0 = 0\) gives good results in practice and yields \(\bar{{\mathbf {G}}} = 0\) in Eq. (17). At each cell \(x_i\), calculate the updated posterior mean function \(\tilde{f}_*^m\) according to Eq. (17) as a highorder GP reconstructor to compute highorder pointwise Riemann state values at \(x_*=x_{i\pm 1/2}\) using each of the candidate substencils \(S_m\). The smoothness indicators (Eq. 42), calculated using the eigensystem from Step (d) in conjunction with the linear weights from Step (c), form the nonlinear weights (Eq. 35). Then take the convex combination according to Eq. (29).
 3.
Calculate fluxes: Solve Riemann problems at cell interfaces using the highorder GP Riemann states in Step 2 as inputs.
 4.
Temporal update: Update the volumeaveraged solutions \(\langle q_i\rangle \) from \(t^n\) to \(t^{n+1}\) using the Godunov fluxes from Step 3.
4 The 1D GP Source Code on GitHub
The full 1D GP source code described in this paper is available at https://github.com/acreyes/GP1D_JSC. The source code is licensed under a Creative Commons Attribution 4.0 International License.
5 Numerical Results
In this section, we present numerical results using the GPWENO reconstructions described in Sect. 2.5 with stencil radii \(R=1,2,3\) (denoted as GPR1, GPR2, GPR3 respectively), applied to the 1D compressible Euler equations and the 1D equations of ideal magnetohydrodynamics (MHD). The GPWENO method will be the default reconstruction scheme hereafter.
We compare the solutions of GPWENO with the fifthorder WENO (referred to as WENOJS in what follows) FVM [26, 55], using the same nonlinear weights in Eq. (35). The only difference between using WENOJS and GPR2 is the use of polynomial based reconstruction and smoothness indicators [55] for WENOJS, and the use of Gaussian process regression with the new GPbased smoothness indicators for GP. All reconstructions are carried out in characteristic variables to minimize unphysical oscillations in the presence of discontinuities. That is, characteristic variables are used in the data vector, \(\mathbf {G}_m\) in Eq. (27). A fourthorder TVD RungeKutta method [54] for temporal updates and the HLLC [38, 63] or Roe [51] Riemann solvers are used throughout.
5.1 Performance Comparison
Shown is the relative time to solution for the four methods considered, all normalized to the GPR2 time
Scheme  Speedup 

GPR1  0.8 
GPR2  1.0 
GPR3  1.4 
WENOJS  0.9 
5.2 1D Smooth Advection
The test considered here involves the passive advection of a Gaussian density profile. We initialize a computational box on [0,1] with periodic boundary conditions. The initial density profile is defined by \(\rho (x) = 1 + e^{100(xx_0)^2}\), with \(x_0=0.5\), with constant velocity, \(u=1\), and pressure, \(P=1/\gamma \). The specific heat ratio is chosen to be \(\gamma =5/3\). The resulting profile is propagated for one period through the boundaries. At \(t=1\), the profile returns to its initial position at \(x=x_0\), any deformation of the initial profile is due to either phase errors or numerical diffusion. We perform this test using a length hyperparameter of \(\ell =0.1\) for stencil radii \(R=1,2,3,4\) and 5, with a fixed Courant number, \(C_{\text {cfl}}=0.8\) and vary the resolution of computational box, with \(N=32,64,128,256\) and 512.
In Fig. 3a, the \(L_1\) error plateaus out at \(\sim 10^{12}\) which is a few orders of magnitude greater than the standard IEEE doubleprecision, \(\sim 10^{16}\). This happens because at high resolution the length hyperparameter, \(\ell \), becomes very large relative to the grid spacing, \(\varDelta \). The covariance matrix, \({\mathbf {C}}\) given in Eq. (22) becomes nearly singular in the regime \(\ell /\varDelta \gg 1\), yielding very large condition numbers for \({\mathbf {C}}\). We find the plateau in the \(L_1\) error occurs for condition numbers, \(\kappa \sim 10^{18}\), corresponding to the point where the errors in inverting \({\mathbf {C}}\) in Eq. (17) begin to dominate. This implies that the choice of floatingpoint precision has an immense impact on the possible \(\ell /\varDelta \), and a proper floatingpoint precision needs to be chosen in such a way that the condition number errors do not dominate. As mentioned in Sect. 2.4, SE suffers from singularity when the size of the dataset grows. The approach that enabled us to produce the results in Fig. 3 is to utilize quadrupleprecision only for the calculation of \(\mathbf {z}^T={\mathbf {T}}_*^{T}{\mathbf {C}}^{1}\) in Eq. (17). Otherwise, the plateau would appear at a much higher \(L_1\) error \(\sim 10^{7}\) with doubleprecision, producing undesirable outcomes for all forms of grid convergence studies. This corresponds to condition numbers \(\kappa \sim 10^8\), and as a point of reference, the breakdown starts to occur for \(\ell /\varDelta >48\) using a GP radius \(R=2\). The weight vector \(\mathbf {z}^T\) needs to be calculated only once, before starting the simulation. It can then be saved and truncated to doubleprecision for use in the actual reconstruction procedure. There are only four related small subroutines that need to be compiled with quadrupleprecision in our code implementation. The overall performance is not affected due to this extra precision handling which is necessary for the purpose of a grid convergence study from the perspective of CFD applications.
The GP’s performance is displayed in Fig. 3b in terms of CPU time per a given error. The GP scheme is computed with different radii, \(R=1,2\) and 3, denoted as GPR1, GPR2 and GPR3, respectively. The GP performances are compared with the three most popular piecewise polynomial methods including the 2ndorder PLM, the 3rdorder PPM, and the 5thorder WENOJS. It is observed that even though GPR2 and WENOJS are equivalent in terms of their nominal 5thorder accuracy, GPR2 always reaches a given error (e.g., the dotted lines show CPU times to reach the error \(\sim 10^{4}\)) 30% faster than WENOJS. Compared to WENOJS, the 7thorder GPR3 is 60% faster, while the 3rdorder GPR1 is 170% slower. The 3rdorder PPM is far slower than WENOJS by 1170%. The 2ndorder PLM is just too slow to be evaluated in this comparison test.
5.3 1D ShuOsher Shock Tube Problem
The second test is the ShuOsher problem [56] to test GP’s shockcapturing capability as well as to see how well GP can resolve smallscale features in the flow. The test gives a good indication of the method’s numerical diffusivity and has been a popular benchmark to demonstrate numerical errors of a given method. In this problem, a (nominally) Mach 3 shock wave propagates into a constant density field with sinusoidal perturbations. As the shock advances, two sets of density features appear behind the shock. One set has the same spatial frequency as the unshocked perturbations, while in the second set the frequency is doubled and follows more closely behind the shock. The primary test of the numerical method is to accurately resolve the dynamics and strengths of the oscillations behind the shock.
The results are shown in Fig. 4. The solutions are calculated at \(t=1.8\) using a resolution of \(N=200\) and are compared to a reference solution resolved on \(N=2056\). It is evident that the GP solution using \(R=3\) provides the least diffusive solution of the methods shown, especially in capturing the amplitude of the postshock oscillations in Fig. 4b. Of the two fifthorder methods, GPR2 and WENOJS, the GP solution has a slightly better amplitude in the postshock oscillations compared to the WENOJS solution, consistent with what is observed in Sect. 5.2 for the smooth advection problem.
The results in Sect. 5.2 suggest that the choice of \(\ell \) should correspond to a length scale characteristic of the flow for optimal performance. Figure 5 compares the postshock features on the same grid for the WENOJS method and the GPR2 method with \(\ell /\varDelta =3, 6\) and 50. Here \(\ell /\varDelta =3\) is roughly a half wavelength of the oscillations, and the GPR2 method clearly gives a much more accurate solution compared to the WENOJS solution. Just as can be seen in Fig. 3c, \(\ell /\varDelta \) much larger than the characteristic length yields a solution much closer to that of WENOJS. From Fig. 3c, we expect that the GP reconstructions in Eq. (27) should approach the WENOJS reconstructions in smooth regions for \(\ell \gg \varDelta \). However, we see that the new GPbased smoothness indicators in Eq. (42) allow the amplitudes near the shock to be better resolved. This reflects a key advantage of the proposed GP method over polynomialbased highorder methods. The additional flexibility in GP, afforded by the kernel hyperparameters, allows its solution accuracy to be tuned to the features that are being resolved. Only at larger values of \(\ell \) does the model become fully constrained by the data in GP, whereas the interpolating polynomials used in a classical WENO are always fully constrained by design. Analogously in the RBF theory, the shape parameters \(\epsilon _j\) plays an important role in the similar context of accuracy and convergence. Several strategies have been studied recently for solving hyperbolic PDEs [3, 21, 29].
5.4 The Sod Shock Tube Test
All of the GPWENO schemes correctly predicts the nonlinear characteristics of the flow including the rarefaction wave, contact discontinuity, and the shock. The solution using GPR2 is very comparable to the WENOJS solution using the same 5point stencil. As expected, the GPR1 solution smears out the most at both the shock and the contact discontinuity, and at the head and tail of the rarefaction. The 7th order GPR3 also successfully demonstrates that its shock solution is physically correct without triggering any unphysical oscillation. Somewhat counterintuitively from the perspective of 1D polynomial schemes, the smallest stencil GPR1 shows the most oscillations near the shock. This happens because the eigendecomposition of the kernel in Eq. (65) used in the calculation the smoothness indicators for Eq. (42) better approximates \(\Vert {\mathbf {f}}^m\Vert _\mathcal {H}^2\) as the size of the stencil increases. All of the GP stencils use the same kernel function, and so \(\Vert {\mathbf {f}}^m\Vert _\mathcal {H}^2\) and the smoothness indicators are calculated from the same eigenspace of the kernel function. Recall that the eigenexpansion of the kernel function in Eq. (63) can contain an infinite number of eigenvalues, and so can \(\Vert {\mathbf {f}}^m\Vert _\mathcal {H}^2\) and the sum in Eq. (42). The finite approximation to the eigensystem and subsequently \(\Vert {\mathbf {f}}^m\Vert _\mathcal {H}^2\) then becomes better as the smallest coefficient \(\alpha _i\) goes to zero. Then for GP, the \(\beta _m\)’s best indicate the smoothness on larger sized stencils, where the sum in Eq. (42) contains more terms in the infinite sum.
5.5 The Einfeldt Strong Rarefaction Test
First described by Einfeldt et al. [14], this problem tests how satisfactorily a code can perform in a lowdensity region in computing physical variables, \(\rho ,u,p,\epsilon \), etc. Among those, the internal energy \(\epsilon =p/(\rho (\gamma 1))\) is particularly difficult to get right due to regions where the density and pressure are very close to zero. Hence the ratio of these two small values amplifies any small errors in both \(\rho \) and p, making the errors in \(\epsilon \) the largest in general [62]. The large errors in \(\epsilon \) are apparent for all schemes shown in Fig. 7, where the error is largest around \(x=0.5\). It can be observed that the amount of departure in \(\epsilon \) from the exact solution (the cyan solid line) decreases as the GP radius R increases. The error in GPR2 is slightly larger for \(\epsilon \) at the center than in WENOJS. However, the peak becomes considerably smaller in amplitude and becomes slightly flattered as R increases.
5.6 BrioWu MHD Shock Tube
Brio and Wu [6] studied an MHD version of Sod’s shock tube problem, which has become an essential test for any MHD code. The test has since uncovered some interesting findings, such as the compound wave [6], as well as the existence of nonunique solutions [64, 65]. The results of this test are shown in Fig. 8.
All of the GP methods are able to satisfactorily capture the MHD structures of the problem. In all methods, including WENOJS, there are some observable oscillations in the postshock regions. Lee in [34] showed that these oscillations arise as a result of the numerical nature of the slowlymoving shock [70] controlled by the strength of the transverse magnetic field. As studied by various researchers [1, 27, 28, 30, 50, 60], there seems no ultimate fix for controlling such unphysical oscillations due to the slowlymoving shock. Quantitatively, the level of oscillations differs in different choices of numerical methods such as reconstruction algorithms and Riemann solvers. We see that all of the GP solutions together with the WENOJS solution feature a comparable level of oscillations. Except for GPR1, all solutions also suffer from a similar type of distortions in u and \(B_y\) in the right going fast rarefaction. This distortion, as well as the oscillations due to the slowlymoving shock, seem to be suppressed in the 3rd order GPR1.
5.7 Ryu and Jones MHD Shock Tubes
Ryu and Jones [52] introduced a large set of MHD shock tube problems as a test of their 1D algorithm, that are now informative to run as a code verification. In what follows, we will refer to the tests as RJ followed by the corresponding figure number from [52] in which the test can be found.
5.7.1 RJ1b Shock Tube
The first of the RJ shock tubes we consider is the RJ1b problem. This test contains a left going fast and slow shock, contact discontinuity as well as a slow and fast rarefaction. In Fig. 9 that all waves are resolved in the schemes considered.
5.7.2 RJ2a Shock Tube
The RJ2a test provides an interesting test due to its initial conditions producing a discontinuity in each of the MHD wave families. The solution contains both fast and slow left and rightmoving magnetoacoustic shocks, left and rightmoving rotational discontinuities and a contact discontinuity. Figure 10 shows that all three of the GP schemes are able to resolve all of these discontinuities. Again, we see that the smallest stencil GPR1 solution contains some oscillations near the shock that are not present in the other GP solutions.
5.7.3 RJ2b Shock Tube
The RJ2b shock tube creates a set of a fast shock, a rotational discontinuity and a slow shock moving to the left away from a contact discontinuity, as well as a fast rarefaction, rotational discontinuity and slow rarefaction all moving to the right. What is of interest is that, since the waves propagate at almost the same speed, at \(t=0.035\) they have still yet to separate much. So, it is important to test a code’s ability to resolve all of the discontinuities despite their close separation. The results of this test for the methods considered are shown in Fig. 11. At the shown resolution of \(N=128\), the contact discontinuity and the slow shock become somewhat smeared together in the GPR1 solution, while they are better resolved by the other methods, even though there are only a couple of grid points distributed over the range of the features.
5.7.4 RJ4a Shock Tube
The RJ4a shock tube yields a fast and slow rarefaction, a contact discontinuity, a slow shock, and of particular interest, the switchon fast shock. The feature of the switchon shock is that the magnetic field is turned on in the region behind the shock. As can be seen in Fig. 12, all of the features including the switchon fast shock are resolved in all methods. We see that GPR1 smears out the solution not only in resolving discontinuous flow regions but also in resolving both fast and slow rarefaction waves.
5.7.5 RJ4b Shock Tube
The RJ4b test is designed to produce only a contact discontinuity and a fast right going switchoff fast rarefaction, where the magnetic field is zero behind the rarefaction. We can see in Fig. 13 that both the contact discontinuity and the switchoff rarefaction features are captured in all of the considered methods.
5.7.6 RJ5b Shock Tube
The RJ5b problem is of interest because it produces a fast compound wave, as opposed to the slow compound wave in the BrioWu problem, in addition to a left and rightgoing slow shock, contact discontinuity and fast rarefaction. At the resolution of \(N=128\) used for the results in Fig. 14, the compound wave and one of the slow shocks are smeared together in all the methods tested.
6 Conclusion
We summarize key novel features of the new highorder GP approach presented in this paper.
The new GP approach utilizes the key idea from statistical theory of GP prediction to produce accurate interpolations of fluid variables in CFD applications. We have developed a new set of numerical strategies of GP for both smooth flows and nonsmooth flows to numerically solve hyperbolic systems of conservation laws.
The GP methods presented here show an extremely fast rate of variable solution accuracy in smooth advection problems by controlling a single parameter, R. Further, the additional flexibility offered by the GP model approach over the fully constrained polynomial based model through the kernel hyperparameter \(\ell \) allows for added tuning of solution accuracy that is not present in traditional polynomial based highorder methods. These parameters allow the GP method to demonstrate variable orders of method accuracy as functions of the size of the GP stencil and the hyperparameter \(\ell \) (see Eq. 21) within a single algorithmic implementation.
The new GPbased smoothness indicators introduced here are used to construct nonlinear weights that give the essentially nonoscillatory property in discontinuous flows. The new smoothness indicators show a significant advantage over traditional WENO schemes in capturing flow features at the grid resolution near discontinuities.
The GP model, by design, can easily be extended to multidimensional GP stencils. Therefore, GP can seamlessly provide a significant algorithmic advantage in solving the multidimensional PDE of CFD. This “dimensional agnosticism” is unique to GP, and not a feature of polynomial methods. We will report our ongoing developments of GP in multiple spatial dimensions in forthcoming papers.
Footnotes
 1.
In the current study R is an integer multiple of \(\varDelta \) for simplicity, which needs not be the case in general.
Notes
Acknowledgements
This work was supported in part by the National Science Foundation under Grant AST0909132; and in part by the U.S. DOE NNSA ASC through the Argonne Institute for Computing in Science under field work proposal 57789. The software used in this work was developed in part by funding from the U.S. DOE NNSAASC and OSOASCR to the Flash Center for Computational Science at the University of Chicago.
References
 1.Arora, M., Roe, P.L.: On postshock oscillations due to shock capturing schemes in unsteady flows. J. Comput. Phys. 130(1), 25–40 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
 2.Attig, N., Gibbon, P., Lippert, T.: Trends in supercomputing: the European path to exascale. Comput. Phys. Commun. 182(9), 2041–2046 (2011)CrossRefGoogle Scholar
 3.Bigoni, C., Hesthaven, J.S.: Adaptive weno methods based on radial basis functions reconstruction. Technical representative. Springer, Berlin (2016)zbMATHGoogle Scholar
 4.Bishop, C.: Pattern recognition and machine learning (information science and statistics), 1st edn. 2006, corr. 2nd printing edn. Springer, New York (2007)Google Scholar
 5.Bond, J., Crittenden, R., Jaffe, A., Knox, L.: Computing challenges of the cosmic microwave background. Comput. Sci. Eng. 1(2), 21–35 (1999)CrossRefGoogle Scholar
 6.Brio, M., Wu, C.C.: An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 75(2), 400–422 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
 7.Buchmüller, P., Helzel, C.: Improved accuracy of highorder WENO finite volume methods on Cartesian grids. J. Sci. Comput. 61(2), 343–368 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
 8.Chen, X., Jung, J.H.: Matrix stability of multiquadric radial basis function methods for hyperbolic equations with uniform centers. J. Sci. Comput. 51(3), 683–702 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
 9.Chen, Y., Gottlieb, S., Heryudono, A., Narayan, A.: A reduced radial basis function method for partial differential equations on irregular domains. J. Sci. Comput. 66(1), 67–90 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
 10.Colella, P., Woodward, P.R.: The piecewise parabolic method (PPM) for gasdynamical simulations. J. Comput. Phys. 54(1), 174–201 (1984)zbMATHCrossRefGoogle Scholar
 11.Cressie, N.: Statistics for Spatial Data. Wiley, Hoboken (2015)zbMATHGoogle Scholar
 12.Dongarra, J.: On the Future of High Performance Computing: How to Think for Peta and Exascale Computing. Hong Kong University of Science and Technology, Hong Kong (2012)Google Scholar
 13.Dongarra, J.J., Meuer, H.W., Simon, H.D., Strohmaier, E.: Recent trends in high performance computing. Birth Numer. Anal. 27, 93 (2010)MathSciNetGoogle Scholar
 14.Einfeldt, B., Munz, C.D., Roe, P.L., Sjögreen, B.: On Godunovtype methods near low densities. J. Comput. Phys. 92(2), 273–295 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
 15.Fasshauer, G.E., Zhang, J.G.: On choosing ”optimal” shape parameters for RBF approximation. Numer. Algorithms 45(1), 345–368 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
 16.Fornberg, B., Wright, G.: Stable computation of multiquadric interpolants for all values of the shape parameter. Comput. Math. Appl. 48(5), 853–867 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
 17.Franke, R.: Scattered data interpolation: tests of some methods. Math. Comput. 38(157), 181–200 (1982)MathSciNetzbMATHGoogle Scholar
 18.Gerolymos, G., Sénéchal, D., Vallet, I.: Veryhighorder WENO schemes. J. Comput. Phys. 228(23), 8481–8524 (2009). https://doi.org/10.1016/j.jcp.2009.07.039 MathSciNetzbMATHCrossRefGoogle Scholar
 19.Godunov, S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. 47(89)(3), 271–306 (1959)MathSciNetzbMATHGoogle Scholar
 20.Gottlieb, D., Shu, C.W.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39(4), 644–668 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
 21.Guo, J., Jung, J.H.: A RBFweno finite volume method for hyperbolic conservation laws with the monotone polynomial interpolation method. Appl. Numer. Math. 112, 27–50 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
 22.Hardy, R.L.: Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 76(8), 1905–1915 (1971)CrossRefGoogle Scholar
 23.Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially nonoscillatory schemes, iii. J. Comput. Phys. 71(2), 231–303 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
 24.Heryudono, A.R., Driscoll, T.A.: Radial basis function interpolation on irregular domain through conformal transplantation. J. Sci. Comput. 44(3), 286–300 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
 25.Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for TimeDependent Problems, vol. 21. Cambridge University Press, Cambridge (2007)zbMATHCrossRefGoogle Scholar
 26.Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
 27.Jin, S., Liu, J.G.: The effects of numerical viscosities: I. slowly moving shocks. J. Comput. Phys. 126(2), 373–389 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
 28.Johnsen, E., Lele, S.: Numerical Errors Generated in Simulations of Slowly Moving shocks, pp. 1–12. Center for Turbulence Research Annual Research Briefs, Stanford (2008)Google Scholar
 29.Jung, J.H., Gottlieb, S., Kim, S.O., Bresten, C.L., Higgs, D.: Recovery of high order accuracy in radial basis function approximations of discontinuous problems. J. Sci. Comput. 45(1), 359–381 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
 30.Karni, S., Čanić, S.: Computations of slowly moving shocks. J. Comput. Phys. 136(1), 132–139 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
 31.Katz, A., Jameson, A.: A comparison of various meshless schemes within a unified algorithm. In: AIAA Paper, vol. 594 (2009)Google Scholar
 32.Keyes, D.E., McInnes, L.C., Woodward, C., Gropp, W., Myra, E., Pernice, M., Bell, J., Brown, J., Clo, A., Connors, J., et al.: Multiphysics simulations challenges and opportunities. Int. J. High Perf. Comput. Appl. 27(1), 4–83 (2013)CrossRefGoogle Scholar
 33.Kolmogorov, A.: Interpolation und Extrapolation von stationären zufalligen Folgen. Izv. Akad. Nauk. SSSR 5, 3–14 (1941)zbMATHGoogle Scholar
 34.Lee, D.: An upwind slope limiter for PPM that preserves monotonicity in magnetohydrodynamics. In: 5th International Conference of Numerical Modeling of Space Plasma Flows (ASTRONUM 2010), vol. 444, p. 236 (2011)Google Scholar
 35.Lee, D.: A solution accurate, efficient and stable unsplit staggered mesh scheme for three dimensional magnetohydrodynamics. J. Comput. Phys. 243, 269–292 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
 36.LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems, vol. 31. Cambridge university press, Cambridge (2002)zbMATHCrossRefGoogle Scholar
 37.LeVeque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations: SteadyState and TimeDependent Problems. SIAM, Bangkok (2007)zbMATHCrossRefGoogle Scholar
 38.Li, S.: An HLLC riemann solver for magnetohydrodynamics. J. Comput. Phys. 203(1), 344–357 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
 39.Liu, X.D., Osher, S., Chan, T.: Weighted essentially nonoscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
 40.Liu, X.Y., Karageorghis, A., Chen, C.: A kansaradial basis function method for elliptic boundary value problems in annular domains. J. Sci. Comput. 65(3), 1240–1269 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
 41.Martel, J.M., Platte, R.B.: Stability of radial basis function methods for convection problems on the circle and sphere. J. Sci. Comput. 69(2), 487–505 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
 42.McCorquodale, P., Colella, P.: A highorder finitevolume method for conservation laws on locally refined grids. Commun. Appl. Math. Comput. Sci. 6(1), 1–25 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
 43.Mignone, A., Tzeferacos, P., Bodo, G.: Highorder conservative finite difference GLMMHD schemes for cellcentered MHD. J. Comput. Phys. 229(17), 5896–5920 (2010). https://doi.org/10.1016/j.jcp.2010.04.013 MathSciNetzbMATHCrossRefGoogle Scholar
 44.Moroney, T.J., Turner, I.W.: A finite volume method based on radial basis functions for twodimensional nonlinear diffusion equations. Appl. Math. Model. 30(10), 1118–1133 (2006)zbMATHCrossRefGoogle Scholar
 45.Moroney, T.J., Turner, I.W.: A threedimensional finite volume method based on radial basis functions for the accurate computational modelling of nonlinear diffusion equations. J. Comput. Phys. 225(2), 1409–1426 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
 46.Morton, K., Sonar, T.: Finite volume methods for hyperbolic conservation laws. Acta Numer. 16(1), 155–238 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
 47.Powell, M.J.: Radial basis funcitionn for multivariable interpolation: a review. In: IMA Conference on Algorithms for the Approximation of Functions and Data, pp. 143–167. RMCS (1985)Google Scholar
 48.Rasmussen, C., Williams, C.: Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning. MIT Press, Cambridge (2005)Google Scholar
 49.Rippa, S.: An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv. Comput. Math. 11(2), 193–210 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
 50.Roberts, T.W.: The behavior of flux difference splitting schemes near slowly moving shock waves. J. Comput. Phys. 90(1), 141–160 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
 51.Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
 52.Ryu, D., Jones, T.: Numerical magnetohydrodynamics in astrophysics: algorithm and tests for onedimensional flow. arXiv:astroph/9404074 (1994)
 53.Shankar, V., Wright, G.B., Kirby, R.M., Fogelson, A.L.: A radial basis function (RBF)finite difference (FD) method for diffusion and reaction–diffusion equations on surfaces. J. Sci. Comput. 63(3), 745–768 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
 54.Shu, C.W.: Totalvariationdiminishing time discretizations. SIAM J. Sci. Stat. Comput. 9(6), 1073–1084 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
 55.Shu, C.W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
 56.Shu, C.W., Osher, S.: Efficient implementation of essentially nonoscillatory shockcapturing schemes, II. J. Comput. Phys. 83(1), 32–78 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
 57.Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27(1), 1–31 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
 58.Sonar, T.: Optimal recovery using thin plate splines in finite volume methods for the numerical solution of hyperbolic conservation laws. IMA J. Numer. Anal. 16(4), 549–581 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
 59.Stein, M.: Interpolation of Spatial Data: Some Theory for Kriging. Springer Series in Statistics Series. Springer, New York (1999)CrossRefGoogle Scholar
 60.Stiriba, Y., Donat, R.: A numerical study of postshock oscillations in slowly moving shock waves. Comput. Math. Appl. 46(5), 719–739 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
 61.Subcommittee, A.: Top ten exascale research challenges. US Department Of Energy Report, 2014 (2014)Google Scholar
 62.Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin (2009)zbMATHCrossRefGoogle Scholar
 63.Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLLriemann solver. Shock Waves 4(1), 25–34 (1994)zbMATHCrossRefGoogle Scholar
 64.Torrilhon, M.: Nonuniform convergence of finite volume schemes for Riemann problems of ideal magnetohydrodynamics. J. Comput. Phys. 192(1), 73–94 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
 65.Torrilhon, M.: Uniqueness conditions for Riemann problems of ideal magnetohydrodynamics. J. Plasma Phys. 69(03), 253–276 (2003)CrossRefGoogle Scholar
 66.Van Leer, B.: Towards the ultimate conservative difference scheme. v. a secondorder sequel to godunov’s method. J. Comput. Phys. 32(1), 101–136 (1979)zbMATHCrossRefGoogle Scholar
 67.Wahba, G., Johnson, D., Gao, F., Gong, J.: Adaptive tuning of numerical weather prediction models: randomized GCV in three and fourdimensional data assimilation. Mon. Weather Rev. 123, 3358–3369 (1995)CrossRefGoogle Scholar
 68.Wendland, H.: Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2010)Google Scholar
 69.Wiener, N.: Extrapolation, interpolation, and smoothing of stationary time series, with engineering applications. Technology Press of the Massachusetts Institute of Technology, Cambridge (1949). ”First published during the war as a classified report to Section D 2, National Defense Research Committee.”; Stationary time seriesGoogle Scholar
 70.Woodward, P., Colella, P.: The numerical simulation of twodimensional fluid flow with strong shocks. J. Comput. Phys. 54(1), 115–173 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
 71.Wright, G.B.: Radial Basis Function Interpolation: Numerical and Analytical Developments. University of Colorado, Boulder (2003)Google Scholar
 72.Wright, G.B., Fornberg, B.: Stable computations with flat radial basis functions using vectorvalued rational approximations. J. Comput. Phys. 331, 137–156 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
 73.Zhang, R., Zhang, M., Shu, C.W.: On the order of accuracy and numerical performance of two classes of finite volume WENO schemes. Commun. Comput. Phys. 9(03), 807–827 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
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