This document describes a Godunov smoothed particle hydrodynamics (SPH) method for simulating geophysical flows over natural terrain. Classical SPH struggles to model flows with steep slopes and resolve shocks accurately. The proposed method overcomes these issues by setting up Riemann problems between interacting particles and using corrected derivative formulas. Boundary conditions are handled by approximating boundaries as piecewise polynomials and using ghost particles. A background mesh is employed for neighbor searching and dynamic load balancing in parallel simulations. Results are presented for cliff collapse and granular jump scenarios.

1. Introduction Issues with Lagrangian Particle Methods Data Structures Results
Godunov Smoothed Particles Hydrodynamics
for Geophysical Flows
Simulations over Natural Terrains
Dinesh Kumar
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2. Introduction Issues with Lagrangian Particle Methods Data Structures Results
State of the Art
Depth Averaged Theory
Unable to model flows on
very steep slopes – a large
part of such hazards ˙
h+ · hv = 0
Incorrect model for ˙
hv + f (hv, g, φ) = g(φ, g, b(x))
interaction with barriers
Details in vertical h is flow-depth,
direction are lost ⇒ v = {vx , vy }T is velocity,
Difficulty in incorporating g is gravity,
physics for erosion / φ is friction angel and
deposition b(x) represents the bed.
Lagrangian particle methods enhanced for shock capturing (e.g.
Godunov SPH) can overcome these problems.
Savage & Hutter, JFM, 10.1017/S0022112089000340, 1989 2/8

3. Introduction Issues with Lagrangian Particle Methods Data Structures Results
Classical SPH can’t resolve shocks
Commonly used artificial viscosity is problem-dependent and
hard to tune a priori. For some probelms it may exceed the
natural viscosity!
Godunov-SPH mj j
Riemann problem setup fi = f W
ρj
between each interacting j
i
pair of particles ρ =ρ
˙ ·v
States projected onto ˙ i
v = mj V ρi , ρj , h p∗ W
local interparticle j
coordinate system p = p(ρ, γ)
Riemann invariants
transformed to global
coordinate system ρ is density, p is pressure,
m is mass and
W is the weight-function
Inutsuka, JCP, 179:238267, 2002 3/8

4. Introduction Issues with Lagrangian Particle Methods Data Structures Results
Accuracy of derivatives
Classical SPH derivatives lose accuracy when there is deficiency
of particles. This introduces errors when solving the mass
balance equation.
Corrected derivatives
Interpolation weights W (xi , xj , h)
W = mj
renormalized to preserve i j
j ρj W (x , x , h)
partition of unity: restoring
mj
accuracy to derivatives when df i j ρj f j − f i W (xi , xj , h)
= mj
there are not enough particles, dx (xj − xi ) W (xi , xj , h)
j ρj
e.g. boundary.
Chen et al, I. J. Num Meth Engr, 46(2):231252, 1999
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5. Introduction Issues with Lagrangian Particle Methods Data Structures Results
Boundary conditions
Traditional methods
Symmetric ghost particles Repulsive boundary forces
Solution
approximate boundary as piecewise-polynomial
uniformly-spaced stationary ghost particles
reflect ghost positions into the domain (only once)
calculate velocities at the ghost-reflections
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6. Introduction Issues with Lagrangian Particle Methods Data Structures Results
Neighbor search and parallelization
Background mesh used for neighbor search and domain
repartition
Mesh resolution determined by support of the weight
function
Domain partitioned in x − y plane only
Dynamic load-balancing, as evolution of flow will create the
computational imbalance between subdomains
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7. Introduction Issues with Lagrangian Particle Methods Data Structures Results
2- D Results
Figure: Cliff Collapse
Figure: Granular Jump 7/8

8. Introduction Issues with Lagrangian Particle Methods Data Structures Results
Thank You
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