| 1. |
- Glaubitz, Jan, et al.
(författare)
-
Energy-Stable Global Radial Basis Function Methods on Summation-By-Parts Form
- 2024
-
Ingår i: Journal of Scientific Computing. - : SPRINGER/PLENUM PUBLISHERS. - 0885-7474 .- 1573-7691. ; 98:1
-
Tidskriftsartikel (refereegranskat)abstract
- Radial basis function methods are powerful tools in numerical analysis and have demonstrated good properties in many different simulations. However, for time-dependent partial differential equations, only a few stability results are known. In particular, if boundary conditions are included, stability issues frequently occur. The question we address in this paper is how provable stability for RBF methods can be obtained. We develop a stability theory for global radial basis function methods using the general framework of summation-by-parts operators often used in the Finite Difference and Finite Element communities. Although we address their practical construction, we restrict the discussion to basic numerical simulations and focus on providing a proof of concept.
|
|
| 2. |
- Glaubitz, Jan, et al.
(författare)
-
Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction
- 2023
-
Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 491
-
Tidskriftsartikel (refereegranskat)abstract
- Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods. © 2023 The Author(s)
|
|
| 3. |
- Glaubitz, Jan, et al.
(författare)
-
Summation-by-Parts Operators for General Function Spaces
- 2023
-
Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 61:2, s. 733-754
-
Tidskriftsartikel (refereegranskat)abstract
- Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions.
|
|
| 4. |
- Glaubitz, Jan, et al.
(författare)
-
Summation-by-parts operators for general function spaces: The second derivative
- 2024
-
Ingår i: Journal of Computational Physics. - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0021-9991 .- 1090-2716. ; 504
-
Tidskriftsartikel (refereegranskat)abstract
- Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for such problems. Conventionally, SBP operators are tailored to the assumption that polynomials accurately approximate the solution, and SBP operators should thus be exact for them. However, this assumption falls short for a range of problems for which other approximation spaces are better suited. We recently addressed this issue and developed a theory for first-derivative SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the innovation of FSBP operators to accommodate second derivatives. The developed second-derivative FSBP operators maintain the desired mimetic properties of existing polynomial SBP operators while allowing for greater flexibility by being applicable to a broader range of function spaces. We establish the existence of these operators and detail a straightforward methodology for constructing them. By exploring various function spaces, including trigonometric, exponential, and radial basis functions, we illustrate the versatility of our approach. The work presented here opens up possibilities for using second-derivative SBP operators based on suitable function spaces, paving the way for a wide range of applications in the future.
|
|
