2:15 PM - 2:30 PM
[MGI30-03] Linear stability analysis of Boussinesq convection with explicit methods and its verification using a high-precision particle scheme
Keywords:Boussinesq convection, Particle methods, Explicit methods, Least squares smoothed particle hydrodynamics (LSSPH) method, Linear stability analysis
Thermal convection is a fundamental mechanism governing planetary dynamics, including planetary mantle convection and atmospheric circulation. In particular, Earth's mantle convection plays a key role in the structure and evolution of its interior because it transports material and heat between the Earth's surface and its deep interior [1].
Mesh-free particle schemes based on explicit methods are effective for thermal convection simulations on a planetary scale. Explicit methods, which enable sequential computations, are more suitable for large-scale parallel computing than implicit methods, which usually solve a massive system of equations [2,3]. Particle schemes can also treat material discontinuity, large deformation, and complex constitutive relations better than grid-based schemes [4]. Therefore, particle schemes are useful for simulating planetary mantle convection that involves partial melting of magma and subducting plates with elastic behaviors. In particular, the smoothed particle hydrodynamics (SPH) method [5,6], one of the particle schemes, is versatile and has been widely used for calculations of star formations in astronomy [e.g., 5,6], giant impacts in planetary sciences [e.g., 7], and dam breaks [8] and landslides [4] in applied engineering problems.
However, the accuracy and stability of the conventional SPH method drastically deteriorate when the particle configuration is disordered [e.g., 9]. Since the discretized models of the conventional SPH method are derived under ideal assumptions that the particle configuration is uniform and the weight function always satisfies the unity condition, their spatial discretization accuracies are reduced to zeroth order or lower when the configuration becomes disordered owing to large deformation [9]. In addition, the particle configuration can easily become irregular due to the sound wave. The sound wave also produces high-frequency fluctuations of the pressure and density fields, often destabilizing the calculation system where the speed of flow is much slower than that of sound, or the Mach number is small [8].
We have developed a highly accurate and stable SPH framework to overcome the above problems [9]. First, we developed a generalized SPH method, namely the least squares SPH (LSSPH) method [9]. This method is based on the idea proposed by Yamamoto and Makino (2017) [10] and maintains at least first-order accuracy under the condition that the ideal assumptions are not fulfilled. Second, we added a density diffusion term [e.g., 8] into the continuity equation. The density diffusion term can rapidly reduce the high-frequency pressure and density fluctuations. On the other hand, although this term is an artificial stabilizing scheme, the proper value of its diffusion parameter has not been discussed and remains unclear for thermal convection systems.
It is also important to accelerate thermal convection simulations. Computing thermal convection for a small Mach number with the SPH method is generally inefficient because the time step restricted by the speed of sound (the CFL condition) is so small. Therefore, applying the reduced speed of sound technique (RSST) [11] is appropriate for a small Mach number system. Since the viscosity is very large for mantle convection, the time interval of the von Neumann condition for the viscosity diffusion is also small. To relax this condition, the variable inertia method (VIM) [3] is valid, where the inertia term whose value is almost zero is increased to a finite value. A technique that combines the RSST and VIM is crucial for mantle convection computations based on explicit methods. On the other hand, it is known that each parameter for RSST and VIM has an upper limit. In particular, when their values are set above certain thresholds, the solution may differ completely from the correct one [12].
In this study, we performed a linear stability analysis of a thermal convection system with the stabilized and accelerating schemes and then investigated the upper limits of their parameters. We considered the Boussinesq convection for simplicity, whereas we included the time derivative term of the density in the continuity equation for employing the RSST. In this presentation, we report the results using the high-precision LSSPH method.
This work was supported by JST SPRING, Grant Number JPMJSP2136.
The references are the same as written in the Japanese abstract.
Mesh-free particle schemes based on explicit methods are effective for thermal convection simulations on a planetary scale. Explicit methods, which enable sequential computations, are more suitable for large-scale parallel computing than implicit methods, which usually solve a massive system of equations [2,3]. Particle schemes can also treat material discontinuity, large deformation, and complex constitutive relations better than grid-based schemes [4]. Therefore, particle schemes are useful for simulating planetary mantle convection that involves partial melting of magma and subducting plates with elastic behaviors. In particular, the smoothed particle hydrodynamics (SPH) method [5,6], one of the particle schemes, is versatile and has been widely used for calculations of star formations in astronomy [e.g., 5,6], giant impacts in planetary sciences [e.g., 7], and dam breaks [8] and landslides [4] in applied engineering problems.
However, the accuracy and stability of the conventional SPH method drastically deteriorate when the particle configuration is disordered [e.g., 9]. Since the discretized models of the conventional SPH method are derived under ideal assumptions that the particle configuration is uniform and the weight function always satisfies the unity condition, their spatial discretization accuracies are reduced to zeroth order or lower when the configuration becomes disordered owing to large deformation [9]. In addition, the particle configuration can easily become irregular due to the sound wave. The sound wave also produces high-frequency fluctuations of the pressure and density fields, often destabilizing the calculation system where the speed of flow is much slower than that of sound, or the Mach number is small [8].
We have developed a highly accurate and stable SPH framework to overcome the above problems [9]. First, we developed a generalized SPH method, namely the least squares SPH (LSSPH) method [9]. This method is based on the idea proposed by Yamamoto and Makino (2017) [10] and maintains at least first-order accuracy under the condition that the ideal assumptions are not fulfilled. Second, we added a density diffusion term [e.g., 8] into the continuity equation. The density diffusion term can rapidly reduce the high-frequency pressure and density fluctuations. On the other hand, although this term is an artificial stabilizing scheme, the proper value of its diffusion parameter has not been discussed and remains unclear for thermal convection systems.
It is also important to accelerate thermal convection simulations. Computing thermal convection for a small Mach number with the SPH method is generally inefficient because the time step restricted by the speed of sound (the CFL condition) is so small. Therefore, applying the reduced speed of sound technique (RSST) [11] is appropriate for a small Mach number system. Since the viscosity is very large for mantle convection, the time interval of the von Neumann condition for the viscosity diffusion is also small. To relax this condition, the variable inertia method (VIM) [3] is valid, where the inertia term whose value is almost zero is increased to a finite value. A technique that combines the RSST and VIM is crucial for mantle convection computations based on explicit methods. On the other hand, it is known that each parameter for RSST and VIM has an upper limit. In particular, when their values are set above certain thresholds, the solution may differ completely from the correct one [12].
In this study, we performed a linear stability analysis of a thermal convection system with the stabilized and accelerating schemes and then investigated the upper limits of their parameters. We considered the Boussinesq convection for simplicity, whereas we included the time derivative term of the density in the continuity equation for employing the RSST. In this presentation, we report the results using the high-precision LSSPH method.
This work was supported by JST SPRING, Grant Number JPMJSP2136.
The references are the same as written in the Japanese abstract.