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In this paper we generalize results regarding the order of accuracy of finite difference operators on summation-by-parts (SBP) form, previously known to hold on uniform grids, to grids with arbitrary point distributions near domain boundaries. We give a definite proof that the order of accuracy in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. Additionally, we prove that if the order of accuracy in the interior is precisely twice that of the boundary, then the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid point distribution near the domain boundaries.

We construct fully discrete stable and accurate numerical schemes for solving partial differential equations posed on non-simply connected spatial domains. The schemes are constructed using summation-by-parts operators in combination with a weak imposition of initial and boundary conditions using the simultaneous approximation term technique. In the theoretical part, we consider the two-dimensional constant coefficient advection equation posed on a rectangular spatial domain with a hole. We construct the new scheme and study well-posedness and stability. Once the theoretical development is done, the technique is extended to more complex non-simply connected geometries. Numerical experiments corroborate the theoretical results and show the applicability of the new approach and its advantages over the standard multiblock technique. Finally, an application using the linearized Euler equations for sound propagation is presented.

Corrigendum to “On the relation between conservation and dual consistency for summation-by-parts schemes”[J. Comput. Phys. 344 (2017) 437–439]

This is a short response to some statements by Wang and Kreiss in “Convergence of Summation-by-Parts Finite Difference Methods for the Wave equation” (Wang and Kreiss in J Sci Comput 71(1):219–245, 2017) that questions results in our paper “On the order of accuracy for difference approximations of initial-boundary value problems” (Svard and Nordstrom in J Comput Phys 218(1):333–352, 2006). We show that our results still stand.

The two dimensional advection–diffusion equation in a stochastically varyinggeometry is considered. The varying domain is transformed into a fixed one andthe numerical solution is computed using a high-order finite difference formulationon summation-by-parts form with weakly imposed boundary conditions. Statistics ofthe solution are computed non-intrusively using quadrature rules given by the probabilitydensity function of the random variable. As a quality control, we prove that thecontinuous problem is strongly well-posed, that the semi-discrete problem is stronglystable and verify the accuracy of the scheme. The technique is applied to a heat transferproblem in incompressible flow. Statistical properties such as confidence intervals andvariance of the solution in terms of two functionals are computed and discussed. Weshow that there is a decreasing sensitivity to geometric uncertainty as we graduallylower the frequency and amplitude of the randomness. The results are less sensitiveto variations in the correlation length of the geometry. Funding agencies: European Union [ACP3-GA-2013-605036]

A detailed account of the stability and accuracy properties of the SBP-SAT technique for numerical time integration is presented. We show how the technique can be used to formulate both global and multi-stage methods with high order of accuracy for both stiff and non-stiff problems. Linear and non- linear stability results, including A-stability, L-stability and B-stability are proven using the energy method for general initial value problems. Numerical experiments corroborate the theoretical properties.

In this paper we derive new far-field boundary conditions for the time-dependent Navier-Stokes and Euler equations in two space dimensions. The new boundary conditions are derived by simultaneously considering well-posedness of both the primal and dual problems. We moreover require that the boundary conditions for the primal and dual Navier-Stokes equations converge to well-posed boundary conditions for the primal and dual Euler equations. We perform computations with a high-order finite difference scheme on summation-by-parts form with the new boundary conditions imposed weakly by the simultaneous approximation term. We prove that the scheme is both energy stable and dual consistent and show numerically that both linear and non-linear integral functionals become superconvergent.

The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion. In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square-roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables, resulting in an unambiguous problem formulation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. For certain stochastic basis functions, the proposed method can be made more effective and well-conditioned. This leads to increased robustness for both choices of variables. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy.

A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate difference schemes using the Summation-By-Parts operators with weak boundary and interface conditions applied to the Hodgkin-Huxley equations. This work is the first demonstrating high accuracy for that equation. Several boundary conditions are considered including the non-standard one accounting for the soma presence, which is characterized by its own partial differential equation. Well-posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions. Gains in terms of CPU times are observed when high-order operators are used, demonstrating the advantage of the high-order schemes for simulating potential propagation in large neuronal trees.

The stochastic Galerkin and collocation methods are used to solve an advection–diffusion equation with uncertain and spatially varying viscosity. We investigate well-posedness, monotonicity and stability for the extended system resulting from the Galerkin projection of the advection–diffusion equation onto the stochastic basis functions. High-order summation-by-parts operators and weak imposition of boundary conditions are used to prove stability of the semi-discrete system. It is essential that the eigenvalues of the resulting viscosity matrix of the stochastic Galerkin system are positive and we investigate conditions for this to hold. When the viscosity matrix is diagonalizable, stochastic Galerkin and stochastic collocation are similar in terms of computational cost, and for some cases the accuracy is higher for stochastic Galerkin provided that monotonicity requirements are met. We also investigate the total spatial operator of the semi-discretized system and its impact on the convergence to steady-state.

Corrigendum to “Summation by parts operators for finite difference approximations of second derivatives” [J.Comput.Phys.199 (2004) 503–540]

A new method for data relaxation based on weak imposition of external data is introduced. The technique is simple, easy to implement, and the resulting numerical scheme is unconditionally stable. Numerical experiments show that the error growth naturally present in long term simulations can be prevented by using the new technique.

We present an ecient, implicit-explicit numerical method for wave propagation insolids containing uid-lled cracks, motivated by applications in geophysical imaging of fracturedoil/gas reservoirs and aquifers, volcanology, and mechanical engineering. We couple the elastic waveequation in the solid to an approximation of the linearized, compressible Navier{Stokes equationsin curved and possibly branching cracks. The approximate uid model, similar to the widely usedlubrication model but accounting for uid inertia and compressibility, exploits the narrowness of thecrack relative to wavelengths of interest. The governing equations are spatially discretized usinghigh-order summation-by-parts nite dierence operators and the uid-solid coupling conditions areweakly enforced, leading to a provably stable scheme. Stiness of the semidiscrete equations can arisefrom the enforcement of coupling conditions, uid compressibility, and diusion operators requiredto capture viscous boundary layers near the crack walls. An implicit-explicit Runge{Kutta scheme isused for time stepping, and the entire system of equations can be advanced in time with high-orderaccuracy using the maximum stable time step determined solely by the standard CFL restriction forwave propagation, irrespective of the crack geometry and uid viscosity. The uid approximationleads to a sparse block structure for the implicit system, such that the additional computationalcost of the uid is small relative to the explicit elastic update. Convergence tests verify highorderaccuracy; additional simulations demonstrate applicability of the method to studies of wavepropagation in and around branching hydraulic fractures. Funding agencies: Baker Hughes; Stanford Natural Gas Initiative; Chevron fellowship in the Department of Geophysics at Stanford University

We present a procedure for constructing Summation-by-Parts operators with minimal dispersion error both near and far from numerical interfaces. Examples of such operators are constructed and compared with a higher order non-optimised Summation-by-Parts operator. Experiments show that the optimised operators are superior for wave propagation and turbulent flows involving large wavenumbers, long solution times and large ranges of resolution scales.

For wave propagation over distances of many wavelengths, high-order finite difference methods on staggered grids are widely used due to their excellent dispersion properties. However, the enforcement of boundary conditions in a stable manner and treatment of interface problems with discontinuous coefficients usually pose many challenges. In this work, we construct a provably stable and high-order-accurate finite difference method on staggered grids that can be applied to a broad class of boundary and interface problems. The staggered grid difference operators are in summation-by-parts form and when combined with a weak enforcement of the boundary conditions, lead to an energy stable method on multiblock grids. The general applicability of the method is demonstrated by simulating an explosive acoustic source, generating waves reflecting against a free surface and material discontinuity. Funding agencies: Department of Geophysics at Stanford University

Incompressible Navier-Stokes solvers based on the projection method often require an expensive numerical solution of a Poisson equation for a pressure-like variable. This often involves linear system solvers based on iterative and multigrid methods which may limit the ability to scale to large numbers of processors. The artificial compressibility method (ACM) [6] introduces a time derivative of the pressure into the incompressible form of the continuity equation creating a coupled closed hyperbolic system that does not require a Poisson equation solution and allows for explicit time-marching and localized stencil numerical methods. Such a scheme should theoretically scale well on large numbers of CPUs, GPU'€™s, or hybrid CPU-GPU architectures. The original ACM was only valid for steady flows and dual-time stepping was often used for time-accurate simulations. Recently, Clausen [7] has proposed the entropically damped artificial compressibility (EDAC) method which is applicable to both steady and unsteady flows without the need for dual-time stepping. The EDAC scheme was successfully tested with both a finite-difference MacCormack'€™s method for the two-dimensional lid driven cavity and periodic double shear layer problem and a finite-element method for flow over a square cylinder, with scaling studies on the latter to large numbers of processors. In this study, we discretize the EDAC formulation with a new optimized high-order centered finite-difference scheme and an explicit fourth-order Runge-€“Kutta method. This is combined with an immersed boundary method to efficiently treat complex geometries and a new robust outflow boundary condition to enable higher Reynolds number simulations on truncated domains. Validation studies for the Taylor-€“Green Vortex problem and the lid driven cavity problem in both 2D and 3D are presented. An eddy viscosity subgrid-scale model is used to enable large eddy simulations for the 3D cases. Finally, an application to flow over a sphere is presented to highlight the boundary condition and performance comparisons to a traditional incompressible Navier-€“Stokes solver is shown for the 3D lid driven cavity. Overall, the combined EDAC formulation and discretization is shown to be both effective and affordable. Funding agencies: Rosenblatt Chair within the faculty of Mechanical Engineering; Zeff Fellowship Trust

We examine the long time error behavior of discontinuous Galerkin spectral element approximations to hyperbolic equations. We show that the choice of numerical flux at interior element boundaries affects the growth rate and asymptotic value of the error. Using the upwind flux, the error reaches the asymptotic value faster, and to a lower value than a central flux gives, especially for low resolution computations. The differences in the error caused by the numerical flux choice decrease as the solution becomes better resolved.

We develop a stable and high-order accurate finite difference method for problems in earthquake rupture dynamics in complex geometries with multiple faults. The bulk material is an isotropic elastic solid cut by pre-existing fault interfaces that accommodate relative motion of the material on the two sides. The fields across the interfaces are related through friction laws which depend on the sliding velocity, tractions acting on the interface, and state variables which evolve according to ordinary differential equations involving local fields. The method is based on summation-by-parts finite difference operators with irregular geometries handled through coordinate transforms and multi-block meshes. Boundary conditions as well as block interface conditions (whether frictional or otherwise) are enforced weakly through the simultaneous approximation term method, resulting in a provably stable discretization. The theoretical accuracy and stability results are confirmed with the method of manufactured solutions. The practical benefits of the new methodology are illustrated in a simulation of a subduction zone megathrust earthquake, a challenging application problem involving complex free-surface topography, nonplanar faults, and varying material properties.

Simulations of the interaction between a vortex and a NACA0012 airfoil are performed with a stable, high-order accurate (in space and time), multi-block finite difference solver for the compressible Navier–Stokes equations. We begin by computing a benchmark test case to validate the code. Next, the flow with steady inflow conditions are computed on several different grids. The resolution of the boundary layer as well as the amount of the artificial dissipation is studied to establish the necessary resolution requirements. We propose an accuracy test based on the weak imposition of the boundary conditions that does not require a grid refinement. Finally, we compute the vortex–airfoil interaction and calculate the lift and drag coefficients. It is shown that the viscous terms add the effect of detailed small scale structures to the lift and drag coefficients. Original Publication:Magnus Svärd, Johan Lundberg and Jan Nordström, A computational study of vortex-airfoil interaction using high-order finite difference methods, 2010, Computers & Fluids, (39), 1267-1274.http://dx.doi.org/10.1016/j.compfluid.2010.03.009Copyright: Elsevier Science B.V., Amsterdam.http://www.elsevier.com/

The Burgers' equation with uncertain initial and boundary conditions is approximated using a Polynomial Chaos Expansion (PCE) approach where the solution is represented as a series of stochastic, orthogonal polynomials. The resulting truncated PCE system is solved using a novel numerical discretization method based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. The resulting PCE solution yields an accurate quantitative description of the stochastic evolution of the system, provided that appropriate boundary conditions are available. The specification of the boundary data is shown to inuence the solution; we will discuss the problematic implications of the lack of precisely characterized boundary data and possible ways of imposing stable and accurate boundary conditions.

Investigation of acceleration effects on missile aerodynamics using computational fluid dynamics

Node-centred edge-based finite volume approximations are very common in computational fluid dynamics since they are assumed to run on structured, unstructured and even on mixed grids. We analyse the accuracy properties of both first and second derivative approximations and conclude that these schemes cannot be used on arbitrary grids as is often assumed. For the Euler equations first-order accuracy can be obtained if care is taken when constructing the grid. For the Navier-Stokes equations, the grid restrictions are so severe that these finite volume schemes have little advantage over structured finite difference schemes. Our theoretical results are verified through extensive computations.

A stable wall boundary procedure is derived for the discretized compressible Navier-Stokes equations. The procedure leads to an energy estimate for the linearized equations. We discretize the equations using high-order accurate finite difference summation-by-parts (SBP) operators. The boundary conditions are imposed weakly with penalty terms. We prove linear stability for the scheme including the wall boundary conditions. The penalty imposition of the boundary conditions is tested for the flow around a circular cylinder at Ma=0.1 and Re=100. We demonstrate the robustness of the SBP-SAT technique by imposing incompatible initial data and show the behavior of the boundary condition implementation. Using the errors at the wall we show that higher convergence rates are obtained for the high-order schemes. We compute the vortex shedding from a circular cylinder and obtain good agreement with previously published (computational and experimental) results for lift, drag and the Strouhal number. We use our results to compare the computational time for a given for a accuracy and show the superior efficiency of the 5th-order scheme.

A novel time integration approach is explored for unsteady flow computations. It is a multi-block formulation in time where one solves for all time levels within a block simultaneously. The time discretization within a block is based on the summation-by-parts (SBP) technique in time combined with the simultaneous-approximation-term (SAT) technique for imposing the initial condition. The approach is implicit, unconditionally stable and can be made high order accurate in time. The implicit system is solved by a dual time stepping technique. The technique has been implemented in a flow solver for unstructured grids and applied to an unsteady flow problem with vortex shedding over a cylinder. Four time integration approaches being 2nd to 5th order accurate in time are evaluated and compared to the conventional 2nd order backward difference (BDF2) method and a 4th order diagonally implicit Runge-Kutta scheme (ESDIRK64). The obtained orders of accuracy are higher than expected and correspond to the accuracy in the interior of the blocks, up to 8th order accuracy is obtained. The influence on the accuracy from the size of the time blocks is small. Smaller blocks are computationally more efficient though, and the efficiency increases with increased accuracy of the SBP operator and reduced size of time steps. The most accurate scheme, with a small time step and block size, is approximately as efficient as the ESDIRK64 scheme. There is a significant potential for improvements ranging from convergence acceleration techniques in dual time, alternative initialization of time blocks, and by introducing smaller time blocks based on alternative SBP operators.

We develop summation-by-parts operators with minimal dispersion errors both near and far from boundaries and interfaces. Such operators are superior to classical stencils for problems involving high frequency waves or multi-frequency solutions over long time intervals with a relatively coarse spatial mesh. This is demonstrated by solving the Taylor-Green vortex flow with optimised and classical operators both in a purely periodic setting as well as in the presence of numerical interfaces.

The accurate and fast prediction of potential propagation in neuronal networks is of prime importance in neurosciences. This work develops a novel structure-preserving model reduction technique to address this problem based on Galerkin projection and nonnegative operator approximation. It is first shown that the corresponding reduced-order model is guaranteed to be energy stable, thanks to both the structure-preserving approach that constructs a distinct reduced-order basis for each cable in the network and the preservation of nonnegativity. Furthermore, a posteriori error estimates are provided, showing that the model reduction error can be bounded and controlled. Finally, the application to the model reduction of a large-scale neuronal network underlines the capability of the proposed approach to accurately predict the potential propagation in such networks while leading to important speedups.

A lateral boundary treatment using summation-by-parts operators and simultaneous approximation terms is introduced. The method is similar to Davies relaxation technique used in the weather prediction community and have similar areas of application, but is also provably stable. In this paper, it is shown how this technique can be applied to the shallow water equations, and that it reduces the errors in the computational domain. Funding agencies: Swedish e-science Research Center (SeRC)

We present a well-posed stochastic Galerkin formulation of the incompressible Navier–Stokes equations with uncertainty in model parameters or the initial and boundary conditions. The stochastic Galerkin method involves representation of the solution through generalized polynomial chaos expansion and projection of the governing equations onto stochastic basis functions, resulting in an extended system of equations. A relatively low-order generalized polynomial chaos expansion is sufficient to capture the stochastic solution for the problem considered. We derive boundary conditions for the continuous form of the stochastic Galerkin formulation of the velocity and pressure equations. The resulting problem formulation leads to an energy estimate for the divergence. With suitable boundary data on the pressure and velocity, the energy estimate implies zero divergence of the velocity field. Based on the analysis of the continuous equations, we present a semi-discretized system where the spatial derivatives are approximated using finite difference operators with a summation-by-parts property. With a suitable choice of dissipative boundary conditions imposed weakly through penalty terms, the semi-discrete scheme is shown to be stable. Numerical experiments in the laminar flow regime corroborate the theoretical results and we obtain high-order accurate results for the solution variables and the velocity divergence converges to zero as the mesh is refined. Funding agencies: SUPRI-B at Stanford University; U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research [DE-SC0006402]; Uni Research, Norway

The advection equation is studied in a completely general two domain setting with different wave-speeds and a time-independent jump-condition at the interface separating the domains. Well-posedness and conservation criteria are derived for the initial-boundary-value problem. The equations are semi-discretized using a finite difference method on Summation-By-Part (SBP) form. The relation between the stability and conservation properties of the approximation are studied when the boundary and interface conditions are weakly imposed by the Simultaneous-Approximation-Term (SAT) procedure. Numerical simulations corroborate the theoretical findings.

In this paper, we present a stable hybrid scheme for viscous problems. The hybrid method combines the unstructured finite volume method with high-order finite difference methods on complex geometries. The coupling procedure between the two numerical methods is based on energy estimates and stable interface conditions are constructed. Numerical calculations show that the hybrid method is efficient and accurate.

We construct a stable high-order finite difference scheme for the compressible Navier-Stokes equations, that satisfy an energy estimate. The equations are discretized with high-order accurate finite difference methods that satisfy a Summation-By-Parts rule. The boundary conditions are imposed with penalty terms known as the Simultaneous Approximation Term technique. The main result is a stability proof for the full three-dimensional Navier-Stokes equations, including the boundary conditions. We show the theoretical third-, fourth-, and fifth-order convergence rate, for a viscous shock, where the analytic solution is known. We demonstrate the stability and discuss the non-reflecting properties of the outflow conditions for a vortex in free space. Furthermore, we compute the three-dimensional vortex shedding behind a circular cylinder in an oblique free stream for Mach number 0.5 and Reynolds number 500.

New sets of boundary conditions for the velocity-pressure formulation of the incompressible Navier-Stokes equations are derived. The boundary conditions have the same form on both inflow and outflow boundaries and lead to a divergence free solution. Moreover, the specific form of the boundary condition makes it possible derive a symmetric positive definite equation system for the internal pressure. Numerical experiments support the theoretical conclusions.

A high-order accurate finite difference scheme is used to perform numerical studies on the benefit of high-order methods. The main advantage of the present technique is the possibility to prove stability for the linearized Euler equations on a multi-block domain, including the boundary conditions. The result is a robust high-order scheme for realistic applications. Convergence studies are presented, verifying design order of accuracy and the superior efficiency of high-order methods for applications dominated by wave propagation. Furthermore, numerical computations of a more complex problem, a vortex-airfoil interaction, show that high-order methods are necessary to capture the significant flow features for transient problems and realistic grid resolutions. This methodology is easy to parallelize due to the multi-block capability. Indeed, we show that the speedup of our numerical method scales almost linearly with the number of processors.

High order finite difference approximations are derived for the second order wave equation with discontinuous coefficients, on rectangular geometries. The discontinuity is treated by splitting the domain at the discontinuities in a multi block fashion. Each sub-domain is discretized with compact second derivative summation by parts operators and the blocks are patched together to a global domain using the projection method. This guarantees a conservative, strictly stable and high order accurate scheme. The analysis is verified by numerical simulations in one and two spatial dimensions.

Our objective is to derive stable first-, second- and fourth-order artificial dissipation operators for node based finite volume schemes. Of particular interest are general unstructured grids where the strength of the finite volume method is fully utilised.A commonly used finite volume approximation of the Laplacian will be the basis in the construction of the artificial dissipation. Both a homogeneous dissipation acting in all directions with equal strength and a modification that allows different amount of dissipation in different directions are derived. Stability and accuracy of the new operators are proved and the theoretical results are supported by numerical computations.

Finite difference approximations of the second derivative in space appearing in, parabolic, incompletely parabolic systems of, and 2nd-order hyperbolic, partial differential equations are considered. If the solution is pointwise bounded, we prove that finite difference approximations of those classes of equations can be closed with two orders less accuracy at the boundary without reducing the global order of accuracy.This result is generalised to initial-boundary value problems with an mth-order principal part. Then, the boundary accuracy can be lowered m orders.Further, it is shown that schemes using summation-by-parts operators that approximate second derivatives are pointwise bounded. Linear and nonlinear computations, including the two-dimensional Navier-Stokes equations, corroborate the theoretical results.

Artificial dissipation terms for finite difference approximations of linear hyperbolic problems with variable coefficients are determined such that an energy estimate and strict stability is obtained. Both conservative and non-conservative approximations are considered. The dissipation terms are computed such that there is no loss of accuracy

We discuss how to combine the node based unstructured finite volume method widely used to handle complex geometries and nonlinear phenomena with very efficient high order finite difference methods suitable for wave propagation dominated problems. This fully coupled numerical procedure reflects the coupled character of the sound generation and propagation problem. The coupling procedure is based on energy estimates and stability can be guaranteed. Numerical experiments using finite difference methods that shed light on the theoretical results are performed. To cite this article: J. Nordstrom, J. Gong, C. R. Mecanique 333 (2005).

In this article we propose a general procedure that allows us to determine both the number and type of boundary conditions for time dependent partial differential equations. With those, well-posedness can be proven for a general initial-boundary value problem. The procedure is exemplified on the linearized Navier--Stokes equations in two and three space dimensions on a general domain.

Finite difference operators approximating second derivatives and satisfying a summation by parts rule have been derived for the fourth, sixth and eighth order case by using the symbolic mathematics software Maple. The operators are based on the same norms as the corresponding approximations of the first derivative, which makes the construction of stable approximations to general parabolic problems straightforward. The error analysis shows that the second derivative approximation can be closed at the boundaries with an approximation two orders less accurate than the internal scheme, and still preserve the internal accuracy.

The unstructured node centered finite volume method is analyzed and it is shown that it can be interpreted in the framework of summation by parts operators. It is also shown that introducing boundary conditions weakly produces strictly stable formulations. Numerical experiments corroborate the analysis.

Electromagnetic wave propagation close to a material discontinuity is simulated by using summation by part operators of second, fourth and sixth order accuracy. The interface conditions at the discontinuity are imposed by the simultaneous approximation term procedure. Stability is shown and the order of accuracy is verified numerically.

The often-used practice of extrapolating all variables at a subsonic, outflow boundary is investigated. For steady state calculations, we show that the L2 error in a subdomain of fixed size decreases with the distance to the far field boundary. Thus, error reduction can be obtained by expanding the size of the computational domain. Numerical experiments using the Euler equations corroborate the theoretical prediction.

We study hyperbolic problems with uncertain stochastically varying geometries. Our aim is to investigate how the stochastically varying uncertainty in the geometry affects the solution of the partial differential equation in terms of the mean and variance of the solution. The problem considered is the two dimensional advection equation on a general domain, which is transformed using curvilinear coordinates to a unit square. The numerical solution is computed using a high order finite difference formulation on summation-by-parts form with weakly imposed boundary conditions. The statistics of the solution are computed nonintrusively using quadrature rules given by the probability density function of the random variable. We prove that the continuous problem is strongly well-posed and that the semi-discrete problem is strongly stable. Numerical calculations using the method of manufactured solution verify the accuracy of the scheme and the statistical properties of the solution are discussed.

A characterisation theorem for best uniform wavenumber approximations by central difference schemes is presented. A central difference stencil is derived based on the theorem and is compared with dispersion relation preserving schemes and with classical central differences for a relevant test problem.