We present a high-order difference method for problems in elastodynamics involving the interactio... more We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We restrict our attention to two-dimensional antiplane problems involving deformation in only one direction. Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity across it are of a form closely related to those used in earthquake rupture models and other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators and weak enforcement of boundary and interface conditions, a strictly stable method is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated in terms of characteristic variables, as opposed to the physical variables in terms of which they are more naturally stated, the semi-discretized system of equations can become extremely stiff, preventing efficient solution using explicit time integrators.
We study an incompletely parabolic system in three space dimensions with stochastic boundary and ... more We study an incompletely parabolic system in three space dimensions with stochastic boundary and initial data. We show how the variance of the solution can be manipulated by the boundary conditions, while keeping the mean value of the solution unaffected. Estimates of the variance of the solution is presented both analytically and numerically. We exemplify the technique by applying it to an incompletely parabolic model problem, as well as the one-dimensional compressible Navier-Stokes equations.
Linköping University Electronic Press eBooks, 2011
This paper evaluates the use of the compressible Navier-Stokes equations, with prescribed zero ve... more This paper evaluates the use of the compressible Navier-Stokes equations, with prescribed zero velocities, as a model for heat transfer in solids. In particular in connection with conjugate heat transfer problems. We derive estimates, and show how to choose and scale the coefficients of the energy part in the Navier-Stokes equations, such that the difference between the energy equation and the heat equation is minimal. A rigorous analysis of the physical interface conditions for the conjugate heat transfer problem is performed and energy estimates are derived in non-standard L 2equivalent norms. The numerical schemes are proven energy stable with the physical interface conditions and the stability of the schemes are independent of the order of accuracy. We have performed computations of a conjugate heat transfer problem in two different ways. One where, as traditionally, the heat transfer in the solid is governed by the heat equation. The other where the heat transfer in the solid is governed by the Navier-Stokes equations. The simulations are compared and the numerical results corroborate the theoretical results.
The two dimensional advection-diffusion equation in a stochastically varying geometry is consider... more The two dimensional advection-diffusion equation in a stochastically varying geometry is considered. The varying domain is transformed into a fixed one and the numerical solution is computed using a high-order finite difference formulation on summation-by-parts form with weakly imposed boundary conditions. Statistics of the solution are computed non-intrusively using quadrature rules given by the probability density function of the random variable. As a quality control, we prove that the continuous problem is strongly well-posed, that the semi-discrete problem is strongly stable and verify the accuracy of the scheme. The technique is applied to a heat transfer problem in incompressible flow. Statistical properties such as confidence intervals and variance of the solution in terms of two functionals are computed and discussed. We show that there is a decreasing sensitivity to geometric uncertainty as we gradually lower the frequency and amplitude of the randomness. The results are less sensitive to variations in the correlation length of the geometry.
In this paper we construct well-posed boundary conditions for the compressible Euler and Navier-S... more In this paper we construct well-posed boundary conditions for the compressible Euler and Navier-Stokes equations in two space dimensions. When also considering the dual equations, we show how to construct the boundary conditions so that both the primal and dual problems are well-posed. By considering the primal and dual problems simultaneously, we construct energy stable and dual consistent finite difference schemes on summation-byparts form with weak imposition of the boundary conditions. According to linear theory, the stable and dual consistent discretization can be used to compute linear integral functionals from the solution at a superconvergent rate. Here we evaluate numerically the superconvergence property for the non-linear Euler and Navier-Stokes equations with linear and non-linear integral functionals.
All numerical calculations will fail to provide a reliable answer unless the continuous problem u... more All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact by discussing well-posedness of the most important equations in computational fluid dynamics, namely the time-dependent compressible Navier-Stokes equations. In particular, we will discuss i) how many boundary conditions are required, ii) where to impose them and iii) which form they should have. The procedure is based on the energy method and generalizes the characteristic boundary procedure for the Euler equations to the compressible Navier-Stokes equations. Once the boundary conditions in terms of i-iii) are known, one issue remains; they can be imposed weakly or strongly. The weak and strong imposition is discussed for the continuous case. It will be shown that the weak and strong boundary procedures produce identical solutions and that the boundary conditions are satisfied exactly also in the weak procedure. We conclude by relating the well-posedness results to energy-stability of the numerical approximation. It is shown that the results obtained in the well-posedness analysis for the continuous problem generalizes directly to stability of the discrete problem.
We present a modified formulation of the dual time-stepping technique which makes use of two deri... more We present a modified formulation of the dual time-stepping technique which makes use of two derivatives in pseudo-time. This new technique retains and improves the convergence properties to the stationary solution. When compared with the conventional dual time-stepping, the method with two derivatives reduces the stiffness of the problem and requires fewer iterations for full convergence to steady-state. In the current formulation, these positive effects require that an approximation of the square root of the spatial operator is available and inexpensive.
An intrusive stochastic projection method for two-phase time-dependent flow subject to uncertaint... more An intrusive stochastic projection method for two-phase time-dependent flow subject to uncertainty is presented. Numerical experiments are carried out assuming uncertainty in the location of the physical interface separating the two phases, but the framework generalizes to uncertainty with known distribution in other input data. Uncertainty is represented through a truncated multiwavelet expansion. We assume that the discontinuous features of the solution are restricted to computational subdomains and use a high-order method for the smooth regions coupled weakly through interfaces with a robust shock capturing method for the non-smooth regions. The discretization of the non-smooth region is based on a generalization of the HLL flux, and have many properties in common with its deterministic counterpart. It is simple and robust, and captures the statistics of the shock.
Finite difference operators satisfying the summation-by-parts (SBP) rules can be used to obtain h... more Finite difference operators satisfying the summation-by-parts (SBP) rules can be used to obtain high order accurate, energy stable schemes for time-dependent partial differential equations, when the boundary conditions are imposed weakly by the simultaneous approximation term (SAT). In general, an SBP-SAT discretization is accurate of order p + 1 with an internal accuracy of 2p and a boundary accuracy of p. Despite this, it is shown in this paper that any linear functional computed from the time-dependent solution, will be accurate of order 2p when the boundary terms are imposed in a stable and dual consistent way. The method does not involve the solution of the dual equations, and superconvergent functionals are obtained at no extra computational cost. Four representative model problems are analyzed in terms of convergence and errors, and it is shown in a systematic way how to derive schemes which gives superconvergent functional outputs.
We study the influence of different implementations of no-slip solid wall boundary conditions on ... more We study the influence of different implementations of no-slip solid wall boundary conditions on the convergence to steady-state of the Navier-Stokes equations. The various approaches are investigated using the energy method and an eigenvalue analysis. It is shown that the weak implementation is superior and enhances the convergence to steady-state for coarse meshes. It is also demonstrated that all the stable approaches produce the same convergence rate as the mesh size goes to zero. The numerical results obtained by using a fully nonlinear finite volume solver support the theoretical findings from the linear analysis.
We construct fully discrete stable and accurate numerical schemes for solving partial differentia... more 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.
In the context of coupling hyperbolic problems, the maximum stable time step of an explicit numer... more In the context of coupling hyperbolic problems, the maximum stable time step of an explicit numerical scheme may depend on the design of the coupling procedure. If this is the case, the coupling procedure is sensitive to changes in model parameters independent of the Courant-Friedrichs-Levy condition. This sensitivity can cause artificial stiffness that degrades the performance of a numerical scheme. To overcome this problem, we present a systematic and general procedure for weakly imposing coupling conditions via penalty terms in a provably non-stiff manner. The procedure can be used to construct both energy conservative and dissipative couplings, and the user is given control over the amount of dissipation desired. The resulting formulation is simple to implement and dual consistent. The penalty coefficients take the form of projection matrices based on the coupling conditions. Numerical experiments demonstrate that this procedure results in both optimal spectral radii and superconvergent linear functionals. Mathematics Subject Classification 65M99 1 Introduction Many scientific and engineering problems require numerical methods that couple different wave phenomena. Numerous applications can be found within computational B Jan Nordström
In this paper we derive well-posed boundary conditions for a linear incompletely parabolic system... more In this paper we derive well-posed boundary conditions for a linear incompletely parabolic system of equations, which can be viewed as a model problem for the compressible Navier-Stokes equations. We show a general procedure for the construction of the boundary conditions such that both the primal and dual equations are wellposed. The form of the boundary conditions is chosen such that reduction to first order form with its complications can be avoided. The primal equation is discretized using finite difference operators on summationby-parts form with weak boundary conditions. It is shown that the discretization can be made energy stable, and that energy stability is sufficient for dual consistency. Since reduction to first order form can be avoided, the discretization is significantly simpler compared to a discretization using Dirichlet boundary conditions. We compare the new boundary conditions with standard Dirichlet boundary conditions in terms of rate of convergence, errors and discrete spectra. It is shown that the scheme with the new boundary conditions is not only far simpler, but also has smaller errors, error bounded properties, and highly optimizable eigenvalues, while maintaining all desirable properties of a dual consistent discretization.
This paper deals with conjugate heat transfer problems for the time-dependent compressible Navier... more This paper deals with conjugate heat transfer problems for the time-dependent compressible Navier-Stokes equations. One way to model conjugate heat transfer is to couple the Navier-Stokes equations in the fluid with the heat equation in the solid. This requires two different physics solvers. Another way is to let the Navier-Stokes equations govern the heat transfer in both the solid and in the fluid. This simplifies calculations since the same physics solver can be used everywhere. We show by energy estimates that the continuous problem is well-posed when imposing continuity of temperature and heat fluxes by using a modified L 2-equivalent norm. The equations are discretized using finite difference on summation-by-parts form with boundary-and interface conditions imposed weakly by the simultaneous approximation term. It is proven that the scheme is energy stable in the modified norm for any order of accuracy.
In this paper we address the error growth in time for hyperbolic problems on first order form. Th... more In this paper we address the error growth in time for hyperbolic problems on first order form. The energy method is used to study when an error growth or a fixed error bound is obtained. It is shown that the choice of boundary procedure is a crucial point. Numerical experiments corroborate the theoretical findings.
The use of implicit methods for numerical time integration typically generates very large systems... more The use of implicit methods for numerical time integration typically generates very large systems of equations, often too large to fit in memory. To address this it is necessary to investigate ways to reduce the sizes of the involved linear systems. We describe a domain decomposition approach for the advection-diffusion equation, based on the Summationby-Parts technique in both time and space. The domain is partitioned into non-overlapping subdomains. A linear system consisting only of interface components is isolated by solving independent subdomain-sized problems. The full solution is then computed in terms of the interface components. The Summation-by-Parts technique provides a solid theoretical framework in which we can mimic the continuous energy method, allowing us to prove both stability and invertibility of the scheme. In a numerical study we show that single-domain implementations of Summation-by-Parts based time integration can be improved upon significantly. Using our proposed method we are able to compute solutions for grid resolutions that cannot be handled efficiently using a single-domain formulation. An order of magnitude speed-up is observed, both compared to a single-domain formulation and to explicit Runge-Kutta time integration.
We discuss conservative and stable numerical approximations in summationby-parts form for linear ... more We discuss conservative and stable numerical approximations in summationby-parts form for linear hyperbolic problems with variable coefficients. An extended setting, where the boundary or interface may or may not be included in the grid, is considered. We prove that conservative and stable formulations for variable coefficient problems require a boundary and interface conforming grid and exact numerical mimicking of integration-by-parts. Finally, we comment on how the conclusions from the linear analysis carry over to the nonlinear setting.
Computer Methods in Applied Mechanics and Engineering, May 1, 2013
The stochastic Galerkin and collocation methods are used to solve an advection-diffusion equation... more 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 summationby-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 steadystate.
This paper concerns energy stability on curvilinear grids and its impact on steady-state calulati... more This paper concerns energy stability on curvilinear grids and its impact on steady-state calulations. We have done computations for the Euler equations using fifth order summation-by-parts block and diagonal norm schemes. By imposing the boundary conditions weakly we obtain a fifth order energy-stable scheme. The calculations indicate the significance of energy stability in order to obtain convergence to steady state. Furthermore, the difference operators are improved such that faster convergence to steady state are obtained.
We present a high-order difference method for problems in elastodynamics involving the interactio... more We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We restrict our attention to two-dimensional antiplane problems involving deformation in only one direction. Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity across it are of a form closely related to those used in earthquake rupture models and other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators and weak enforcement of boundary and interface conditions, a strictly stable method is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated in terms of characteristic variables, as opposed to the physical variables in terms of which they are more naturally stated, the semi-discretized system of equations can become extremely stiff, preventing efficient solution using explicit time integrators.
We study an incompletely parabolic system in three space dimensions with stochastic boundary and ... more We study an incompletely parabolic system in three space dimensions with stochastic boundary and initial data. We show how the variance of the solution can be manipulated by the boundary conditions, while keeping the mean value of the solution unaffected. Estimates of the variance of the solution is presented both analytically and numerically. We exemplify the technique by applying it to an incompletely parabolic model problem, as well as the one-dimensional compressible Navier-Stokes equations.
Linköping University Electronic Press eBooks, 2011
This paper evaluates the use of the compressible Navier-Stokes equations, with prescribed zero ve... more This paper evaluates the use of the compressible Navier-Stokes equations, with prescribed zero velocities, as a model for heat transfer in solids. In particular in connection with conjugate heat transfer problems. We derive estimates, and show how to choose and scale the coefficients of the energy part in the Navier-Stokes equations, such that the difference between the energy equation and the heat equation is minimal. A rigorous analysis of the physical interface conditions for the conjugate heat transfer problem is performed and energy estimates are derived in non-standard L 2equivalent norms. The numerical schemes are proven energy stable with the physical interface conditions and the stability of the schemes are independent of the order of accuracy. We have performed computations of a conjugate heat transfer problem in two different ways. One where, as traditionally, the heat transfer in the solid is governed by the heat equation. The other where the heat transfer in the solid is governed by the Navier-Stokes equations. The simulations are compared and the numerical results corroborate the theoretical results.
The two dimensional advection-diffusion equation in a stochastically varying geometry is consider... more The two dimensional advection-diffusion equation in a stochastically varying geometry is considered. The varying domain is transformed into a fixed one and the numerical solution is computed using a high-order finite difference formulation on summation-by-parts form with weakly imposed boundary conditions. Statistics of the solution are computed non-intrusively using quadrature rules given by the probability density function of the random variable. As a quality control, we prove that the continuous problem is strongly well-posed, that the semi-discrete problem is strongly stable and verify the accuracy of the scheme. The technique is applied to a heat transfer problem in incompressible flow. Statistical properties such as confidence intervals and variance of the solution in terms of two functionals are computed and discussed. We show that there is a decreasing sensitivity to geometric uncertainty as we gradually lower the frequency and amplitude of the randomness. The results are less sensitive to variations in the correlation length of the geometry.
In this paper we construct well-posed boundary conditions for the compressible Euler and Navier-S... more In this paper we construct well-posed boundary conditions for the compressible Euler and Navier-Stokes equations in two space dimensions. When also considering the dual equations, we show how to construct the boundary conditions so that both the primal and dual problems are well-posed. By considering the primal and dual problems simultaneously, we construct energy stable and dual consistent finite difference schemes on summation-byparts form with weak imposition of the boundary conditions. According to linear theory, the stable and dual consistent discretization can be used to compute linear integral functionals from the solution at a superconvergent rate. Here we evaluate numerically the superconvergence property for the non-linear Euler and Navier-Stokes equations with linear and non-linear integral functionals.
All numerical calculations will fail to provide a reliable answer unless the continuous problem u... more All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact by discussing well-posedness of the most important equations in computational fluid dynamics, namely the time-dependent compressible Navier-Stokes equations. In particular, we will discuss i) how many boundary conditions are required, ii) where to impose them and iii) which form they should have. The procedure is based on the energy method and generalizes the characteristic boundary procedure for the Euler equations to the compressible Navier-Stokes equations. Once the boundary conditions in terms of i-iii) are known, one issue remains; they can be imposed weakly or strongly. The weak and strong imposition is discussed for the continuous case. It will be shown that the weak and strong boundary procedures produce identical solutions and that the boundary conditions are satisfied exactly also in the weak procedure. We conclude by relating the well-posedness results to energy-stability of the numerical approximation. It is shown that the results obtained in the well-posedness analysis for the continuous problem generalizes directly to stability of the discrete problem.
We present a modified formulation of the dual time-stepping technique which makes use of two deri... more We present a modified formulation of the dual time-stepping technique which makes use of two derivatives in pseudo-time. This new technique retains and improves the convergence properties to the stationary solution. When compared with the conventional dual time-stepping, the method with two derivatives reduces the stiffness of the problem and requires fewer iterations for full convergence to steady-state. In the current formulation, these positive effects require that an approximation of the square root of the spatial operator is available and inexpensive.
An intrusive stochastic projection method for two-phase time-dependent flow subject to uncertaint... more An intrusive stochastic projection method for two-phase time-dependent flow subject to uncertainty is presented. Numerical experiments are carried out assuming uncertainty in the location of the physical interface separating the two phases, but the framework generalizes to uncertainty with known distribution in other input data. Uncertainty is represented through a truncated multiwavelet expansion. We assume that the discontinuous features of the solution are restricted to computational subdomains and use a high-order method for the smooth regions coupled weakly through interfaces with a robust shock capturing method for the non-smooth regions. The discretization of the non-smooth region is based on a generalization of the HLL flux, and have many properties in common with its deterministic counterpart. It is simple and robust, and captures the statistics of the shock.
Finite difference operators satisfying the summation-by-parts (SBP) rules can be used to obtain h... more Finite difference operators satisfying the summation-by-parts (SBP) rules can be used to obtain high order accurate, energy stable schemes for time-dependent partial differential equations, when the boundary conditions are imposed weakly by the simultaneous approximation term (SAT). In general, an SBP-SAT discretization is accurate of order p + 1 with an internal accuracy of 2p and a boundary accuracy of p. Despite this, it is shown in this paper that any linear functional computed from the time-dependent solution, will be accurate of order 2p when the boundary terms are imposed in a stable and dual consistent way. The method does not involve the solution of the dual equations, and superconvergent functionals are obtained at no extra computational cost. Four representative model problems are analyzed in terms of convergence and errors, and it is shown in a systematic way how to derive schemes which gives superconvergent functional outputs.
We study the influence of different implementations of no-slip solid wall boundary conditions on ... more We study the influence of different implementations of no-slip solid wall boundary conditions on the convergence to steady-state of the Navier-Stokes equations. The various approaches are investigated using the energy method and an eigenvalue analysis. It is shown that the weak implementation is superior and enhances the convergence to steady-state for coarse meshes. It is also demonstrated that all the stable approaches produce the same convergence rate as the mesh size goes to zero. The numerical results obtained by using a fully nonlinear finite volume solver support the theoretical findings from the linear analysis.
We construct fully discrete stable and accurate numerical schemes for solving partial differentia... more 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.
In the context of coupling hyperbolic problems, the maximum stable time step of an explicit numer... more In the context of coupling hyperbolic problems, the maximum stable time step of an explicit numerical scheme may depend on the design of the coupling procedure. If this is the case, the coupling procedure is sensitive to changes in model parameters independent of the Courant-Friedrichs-Levy condition. This sensitivity can cause artificial stiffness that degrades the performance of a numerical scheme. To overcome this problem, we present a systematic and general procedure for weakly imposing coupling conditions via penalty terms in a provably non-stiff manner. The procedure can be used to construct both energy conservative and dissipative couplings, and the user is given control over the amount of dissipation desired. The resulting formulation is simple to implement and dual consistent. The penalty coefficients take the form of projection matrices based on the coupling conditions. Numerical experiments demonstrate that this procedure results in both optimal spectral radii and superconvergent linear functionals. Mathematics Subject Classification 65M99 1 Introduction Many scientific and engineering problems require numerical methods that couple different wave phenomena. Numerous applications can be found within computational B Jan Nordström
In this paper we derive well-posed boundary conditions for a linear incompletely parabolic system... more In this paper we derive well-posed boundary conditions for a linear incompletely parabolic system of equations, which can be viewed as a model problem for the compressible Navier-Stokes equations. We show a general procedure for the construction of the boundary conditions such that both the primal and dual equations are wellposed. The form of the boundary conditions is chosen such that reduction to first order form with its complications can be avoided. The primal equation is discretized using finite difference operators on summationby-parts form with weak boundary conditions. It is shown that the discretization can be made energy stable, and that energy stability is sufficient for dual consistency. Since reduction to first order form can be avoided, the discretization is significantly simpler compared to a discretization using Dirichlet boundary conditions. We compare the new boundary conditions with standard Dirichlet boundary conditions in terms of rate of convergence, errors and discrete spectra. It is shown that the scheme with the new boundary conditions is not only far simpler, but also has smaller errors, error bounded properties, and highly optimizable eigenvalues, while maintaining all desirable properties of a dual consistent discretization.
This paper deals with conjugate heat transfer problems for the time-dependent compressible Navier... more This paper deals with conjugate heat transfer problems for the time-dependent compressible Navier-Stokes equations. One way to model conjugate heat transfer is to couple the Navier-Stokes equations in the fluid with the heat equation in the solid. This requires two different physics solvers. Another way is to let the Navier-Stokes equations govern the heat transfer in both the solid and in the fluid. This simplifies calculations since the same physics solver can be used everywhere. We show by energy estimates that the continuous problem is well-posed when imposing continuity of temperature and heat fluxes by using a modified L 2-equivalent norm. The equations are discretized using finite difference on summation-by-parts form with boundary-and interface conditions imposed weakly by the simultaneous approximation term. It is proven that the scheme is energy stable in the modified norm for any order of accuracy.
In this paper we address the error growth in time for hyperbolic problems on first order form. Th... more In this paper we address the error growth in time for hyperbolic problems on first order form. The energy method is used to study when an error growth or a fixed error bound is obtained. It is shown that the choice of boundary procedure is a crucial point. Numerical experiments corroborate the theoretical findings.
The use of implicit methods for numerical time integration typically generates very large systems... more The use of implicit methods for numerical time integration typically generates very large systems of equations, often too large to fit in memory. To address this it is necessary to investigate ways to reduce the sizes of the involved linear systems. We describe a domain decomposition approach for the advection-diffusion equation, based on the Summationby-Parts technique in both time and space. The domain is partitioned into non-overlapping subdomains. A linear system consisting only of interface components is isolated by solving independent subdomain-sized problems. The full solution is then computed in terms of the interface components. The Summation-by-Parts technique provides a solid theoretical framework in which we can mimic the continuous energy method, allowing us to prove both stability and invertibility of the scheme. In a numerical study we show that single-domain implementations of Summation-by-Parts based time integration can be improved upon significantly. Using our proposed method we are able to compute solutions for grid resolutions that cannot be handled efficiently using a single-domain formulation. An order of magnitude speed-up is observed, both compared to a single-domain formulation and to explicit Runge-Kutta time integration.
We discuss conservative and stable numerical approximations in summationby-parts form for linear ... more We discuss conservative and stable numerical approximations in summationby-parts form for linear hyperbolic problems with variable coefficients. An extended setting, where the boundary or interface may or may not be included in the grid, is considered. We prove that conservative and stable formulations for variable coefficient problems require a boundary and interface conforming grid and exact numerical mimicking of integration-by-parts. Finally, we comment on how the conclusions from the linear analysis carry over to the nonlinear setting.
Computer Methods in Applied Mechanics and Engineering, May 1, 2013
The stochastic Galerkin and collocation methods are used to solve an advection-diffusion equation... more 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 summationby-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 steadystate.
This paper concerns energy stability on curvilinear grids and its impact on steady-state calulati... more This paper concerns energy stability on curvilinear grids and its impact on steady-state calulations. We have done computations for the Euler equations using fifth order summation-by-parts block and diagonal norm schemes. By imposing the boundary conditions weakly we obtain a fifth order energy-stable scheme. The calculations indicate the significance of energy stability in order to obtain convergence to steady state. Furthermore, the difference operators are improved such that faster convergence to steady state are obtained.
Uploads
Papers by Jan Nordström