## Final Report Summary - USEFUL (USEFUL SUPERCONVERGENCE EXTRACTION FOR FILTERING OF UNDERRESOLVED SOLUTIONS)

1.1 The objective

The purpose of the project is to construct a mathematically rigorous and computationally efficient explicit filter that can be used for computing nonlinear equations and eventually LES. This will ensure that errors from the numerical approximation are minimized. This requires a thorough investigation of the divided difference estimates for discontinuous Galerkin (DG) methods. The theoretical superconvergence results obtained are the guiding principles in designing some appropriate Smoothness-Increasing Accuracy-Conserving (SIAC) filters that are composed of B-splines with different orders or numbers. These approaches and results are helpful for modelling more engineering-type problems such as wind and tidal energy etc.

1.2 The main work and results

To better understand how to create SIAC filters for different nonlinear equation, we are mainly focused on the divided difference estimates of the DG errors for nonlinear scalar and systems of hyperbolic equations. Since negative-order norm error estimates of divided differences are important tools to derive superconvergent error estimates of the post-processed solutions with some special kernels, both the L2 norm and negative-order norm of the divided difference of the DG error are analysed. Specifically, for the nonlinear scalar and system case of hyperbolic equations,

- the following significant theoretical results are proven:

o By establishing some relations between the spatial and time derivatives of the projection of the DG errors, the L2 norm error estimate of order k+(3-a)/2 are proved for the a-th order divided difference.

o By the duality argument and constructing a suitable dual problem, a superconvergence result of order 2k+(3-a)/2 is obtained for the negative-order norm estimate.

o By the post-processing theory, the superconvergent negative-order norm estimate implies a superconvergent result of the same order for the L2 norm of the post-processed errors.

- the following significant computational results are observed:

o According to the proven superconvergent error estimates, a more compact kernel function is identified. However, this may vary with different nonlinear equations and different polynomials degrees of the discontinuous finite element space.

o For solutions containing discontinuities, the errors of our more compact kernel are less oscillatory than that for the standard full kernel composed of 2k+1 B-splines of order k+1.

1.3 The main difficulties

We would like to emphasize that extension of the post-processing theory of the SIAC filter from the linear case to the nonlinear case is very complicated and technical. Indeed, even for the L2 norm estimates of the divided difference of the DG error for scalar conservation laws, the analysis involves some technicalities, and in particular how to deal with the cross term involving both spatial and temporal derivatives that are required by the DG discretisation operator. This issue, however, is addressed by establishing some relations between spatial and time derivatives of the projection of the DG error together with investigation and application of properties of divided differences. As to the extension to the more general nonlinear system case, we not only need to deal with the divided difference of the projection of the error, but also concentrate on the analysis of the divided difference of the projection error. Thus, we have carefully studied the definition and property of the upwind flux as well as the Gauss-Radau projections. Moreover, based on the local characteristic decomposition approach together with the eigen-structures, the L2 norm estimate of the divided difference of the Jacobi matrix and thus the projection error is proved to be of optimal order of k+1.

1.4 Conclusions

Through the investigation of theoretical and numerical aspects of SIAC filters for nonlinear problems including scalar and system cases, we conclude that

- The divided difference estimates of the DG errors play an important role in obtaining negative-order norm estimates and thus the superconvergent results for post-processed solutions.

- The proven negative-order norm estimates enable us to design some computationally efficient explicit filters to enhance the accuracy of the method with further applications to LES and other engineering problems.

- The proposed filters are more compact and are less oscillatory for computing solutions containing shocks.

1.5 Potential impact resulting from the project

The final results of the project are expected to provide a unified framework regarding numerical analysis and simulation of DG methods for compressible Navier-Stokes equation and in particular to resolve LES solutions using some suitable SIAC filters. Specifically,

- Theoretical and computational insight into the effectiveness of SIAC filter for nonlinear systems of convection-diffusion equations.

- Construction of appropriate filters to resolve solutions of LES with different scales.

- The societal implication of the project is to minimize the mathematical errors in computational simulations that are useful in areas such as climate research, environment protection etc.

To demonstrate the effectiveness of the SIAC filter, attachments have been provided for the nonlinear Euler equation with source terms. These attachments show the numerical errors, convergence orders in L2 and L∞ norm (in Table 1.5.1), pointwise errors (in Figure 1.5.2) as well as order plot (in Figure 1.5.1).

The purpose of the project is to construct a mathematically rigorous and computationally efficient explicit filter that can be used for computing nonlinear equations and eventually LES. This will ensure that errors from the numerical approximation are minimized. This requires a thorough investigation of the divided difference estimates for discontinuous Galerkin (DG) methods. The theoretical superconvergence results obtained are the guiding principles in designing some appropriate Smoothness-Increasing Accuracy-Conserving (SIAC) filters that are composed of B-splines with different orders or numbers. These approaches and results are helpful for modelling more engineering-type problems such as wind and tidal energy etc.

1.2 The main work and results

To better understand how to create SIAC filters for different nonlinear equation, we are mainly focused on the divided difference estimates of the DG errors for nonlinear scalar and systems of hyperbolic equations. Since negative-order norm error estimates of divided differences are important tools to derive superconvergent error estimates of the post-processed solutions with some special kernels, both the L2 norm and negative-order norm of the divided difference of the DG error are analysed. Specifically, for the nonlinear scalar and system case of hyperbolic equations,

- the following significant theoretical results are proven:

o By establishing some relations between the spatial and time derivatives of the projection of the DG errors, the L2 norm error estimate of order k+(3-a)/2 are proved for the a-th order divided difference.

o By the duality argument and constructing a suitable dual problem, a superconvergence result of order 2k+(3-a)/2 is obtained for the negative-order norm estimate.

o By the post-processing theory, the superconvergent negative-order norm estimate implies a superconvergent result of the same order for the L2 norm of the post-processed errors.

- the following significant computational results are observed:

o According to the proven superconvergent error estimates, a more compact kernel function is identified. However, this may vary with different nonlinear equations and different polynomials degrees of the discontinuous finite element space.

o For solutions containing discontinuities, the errors of our more compact kernel are less oscillatory than that for the standard full kernel composed of 2k+1 B-splines of order k+1.

1.3 The main difficulties

We would like to emphasize that extension of the post-processing theory of the SIAC filter from the linear case to the nonlinear case is very complicated and technical. Indeed, even for the L2 norm estimates of the divided difference of the DG error for scalar conservation laws, the analysis involves some technicalities, and in particular how to deal with the cross term involving both spatial and temporal derivatives that are required by the DG discretisation operator. This issue, however, is addressed by establishing some relations between spatial and time derivatives of the projection of the DG error together with investigation and application of properties of divided differences. As to the extension to the more general nonlinear system case, we not only need to deal with the divided difference of the projection of the error, but also concentrate on the analysis of the divided difference of the projection error. Thus, we have carefully studied the definition and property of the upwind flux as well as the Gauss-Radau projections. Moreover, based on the local characteristic decomposition approach together with the eigen-structures, the L2 norm estimate of the divided difference of the Jacobi matrix and thus the projection error is proved to be of optimal order of k+1.

1.4 Conclusions

Through the investigation of theoretical and numerical aspects of SIAC filters for nonlinear problems including scalar and system cases, we conclude that

- The divided difference estimates of the DG errors play an important role in obtaining negative-order norm estimates and thus the superconvergent results for post-processed solutions.

- The proven negative-order norm estimates enable us to design some computationally efficient explicit filters to enhance the accuracy of the method with further applications to LES and other engineering problems.

- The proposed filters are more compact and are less oscillatory for computing solutions containing shocks.

1.5 Potential impact resulting from the project

The final results of the project are expected to provide a unified framework regarding numerical analysis and simulation of DG methods for compressible Navier-Stokes equation and in particular to resolve LES solutions using some suitable SIAC filters. Specifically,

- Theoretical and computational insight into the effectiveness of SIAC filter for nonlinear systems of convection-diffusion equations.

- Construction of appropriate filters to resolve solutions of LES with different scales.

- The societal implication of the project is to minimize the mathematical errors in computational simulations that are useful in areas such as climate research, environment protection etc.

To demonstrate the effectiveness of the SIAC filter, attachments have been provided for the nonlinear Euler equation with source terms. These attachments show the numerical errors, convergence orders in L2 and L∞ norm (in Table 1.5.1), pointwise errors (in Figure 1.5.2) as well as order plot (in Figure 1.5.1).