Description_
This_book_presents_contemporary_computational_methods_for_solving_major_problems_of_current_interest,_from_underwater_sound_scattering_to_space-time_issues_to_aerodynamics._It_describes_the_basic_physics_of_the_associated_wave_phenomena._Surveys_state-of-the-art_developments,_such_as_the_discretization_of_local_absorbing_boundary_conditions_and_the_perfectly_matched_layer_technique._Includes_applications_of_waves_in_compressible_fluids_and_nonlinear_dispersive,_acoustic,_elastic,_and_seismic_waves._Due_to_the_increase_in_computational_power_and_new_discoveries_in_propagation_phenomena_for_linear_and_nonlinear_waves,_the_area_of_computational_wave_propagation_has_become_more_significant_in_recent_years._Exploring_the_latest_developments_in_the_field,_Effective_Computational_Methods_for_Wave_Propagation_presents_several_modern,_valuable_computational_methods_used_to_describe_wave_propagation_phenomena_in_selected_areas_of_physics_and_technology._Featuring_contributions_from_internationally_known_experts,_the_book_is_divided_into_four_parts._It_begins_with_the_simulation_of_nonlinear_dispersive_waves_from_nonlinear_optics_and_the_theory_and_numerical_analysis_of_Boussinesq_systems._The_next_section_focuses_on_computational_approaches,_including_a_finite_element_method_and_parabolic_equation_techniques,_for_mathematical_models_of_underwater_sound_propagation_and_scattering._The_book_then_offers_a_comprehensive_introduction_to_modern_numerical_methods_for_time-dependent_elastic_wave_propagation._The_final_part_supplies_an_overview_of_high-order,_low_diffusion_numerical_methods_for_complex,_compressible_flows_of_aerodynamics._Concentrating_on_physics_and_technology,_this_volume_provides_the_necessary_computational_methods_to_effectively_tackle_the_sources_of_problems_that_involve_some_type_of_wave_motion._

Preface
Waves_are_ubiquitous_in_nature._We_are_all_familiar_with_water_waves,
sound_waves,_electromagnetic_and_seismic_waves._It_is_fair_to_say_that_every
area_of_science_and_technology_has_sources_of_problems_involving_some_type
of_wave_motion._As_a_result,_a_large_arsenal_of_analytical_techniques_has_been
developed_to_describe_and_analyze_linear_and_nonlinear_wave_phenomena._And,
of_course,_the_numerical_simulation_of_wave_motion,_mainly_the_numerical
solution_of_the_partial_differential_equations_of_wave_theory,_has_been_at_the
center_of_attention_of_computational_scientists_since_the_advent_of_the_modern
computer_era.
The_aim_of_the_volume_at_hand_is_to_present_some_modern,_effective_computational
methods_used_to_describe_wave_propagation_phenomena_in_selected
areas_of_current_interest_in_physics_and_technology._One_cannot_hope_to_be
comprehensive_in_such_an_effort._It_was_the_editors’_choice_to_concentrate_in
four_areas,_which_are,_inevitably,_close_to_their_interests_and_expertise:
I._Nonlinear_dispersive_waves.
II._The_Helmholtz_equation_and_its_paraxial_approximations_in_underwater
acoustics.
III._Numerical_methods_in_elastic_wave_propagation.
IV._Waves_in_compressible_flows.
Each_part_consists_of_chapters_(or_articles)_on_specific_topics,_written_by_internationally
known_experts._Part_I_begins_with_two_articles_on_the_simulation
of_nonlinear_dispersive_waves_from_nonlinear_optics._In_Chapter_1,_X.-P.Wang
reviews_the_dynamic_rescaling_technique_and_the_iterative_grid_redistribution
method_for_approximating_singular_(blow-up)_and_near-singular_solutions_of
the_Nonlinear_Schr¨odinger_equation._In_Chapter_2,_G.Fibich_and_S.Tsynkov
consider_the_Nonlinear_Helmholtz_equation_describing_time-harmonic_electromagnetic
waves_in_Kerr_media._They_pose_boundary-value_problems_for_this
type_of_equation_with_nonlocal_artificial_boundary_conditions_at_the_nearand
far-boundaries_of_the_half-space_in_the_direction_of_which_propagation
of_the_wave_mainly_takes_place,_and_radiation_conditions_on_the_transverse
boundaries,_and_they_solve_them_numerically_by_high-order_finite_difference
methods._In_Chapter_3,_V._A._Dougalis_and_D._E._Mitsotakis,_after_reviewing
1
2
the_derivation_and_the_well-posedness_theory_of_a_class_of_Boussinesq_type_systems
that_approximate_the_Euler_equations_of_water_wave_theory_and_describe
two-way_propagation_of_long_waves_of_small_amplitude,_they_investigate_by
analytical_and_numerical_means_(using_fully_discrete_Galerkin-finite_element
methods)_solitary_waves_of_these_systems_and_their_role_in_the_evolution_of
general_solutions.
Part_II_consists_of_four_chapters_on_computational_techniques_for_mathematical
models_of_underwater_sound_propagation_and_scattering._In_Chapter
4,_D._A._Mitsoudis,_N._A._Kampanis_and_V._A._Dougalis_present_a_finite_element
method_for_the_(linear)_Helmholtz_equation_in_a_general_fluid_waveguide
with_range-dependent_layer_topography_and_concentrate_on_the_implementation
and_the_coupling_of_the_finite_element_method_with_DtN_type_nonlocal
boundary_conditions_at_the_inflow_and_outflow_boundaries_of_the_waveguide.
The_next_three_chapters_concern_various_issues_related_to_paraxial_(‘parabolic’)
approximations_to_the_Helmholtz_equation,_that_have_been_successfully_used
to_model_long-range_propagation_of_sound_in_the_sea._First,_D._J._Thomson
and_G._H._Brooke_in_Chapter_5_present_an_overview_of_Parabolic_Equation
(PE)_techniques_in_underwater_acoustics,_including_an_introduction_to
PE-based_matched_field_processing_techniques_for_source_localization_and_for
decomposing_the_acoustic_field_into_its_modal_components._In_the_following
Chapter_6,_V._A._Dougalis,_N._A._Kampanis,_F._Sturm_and_G._E._Zouraris
address_modelling_and_numerical_issues_for_the_PE_and_its_higher-order_wideangle
analogs_in_waveguides_with_range-dependent_interfaces_and_bottoms_in
axisymmetric_and_fully_3D_environments,_when_range-dependent_changes_of
variable_are_used_to_make_the_layers_horizontal._Finally,_in_Chapter_7,_G._H.
Brooke,_D._J._Thomson_and_the_late_T._W._Dawson,_after_reviewing_various
nonlocal_boundary_conditions_that_are_applied_at_interfaces_between_the_computational
domain_and_an_external_half_space_(transverse_to_the_direction_of
propagation)_in_the_case_of_the_PE,_they_study_a_particular_half_space_with_linear
squared_index_of_refraction_and_show_how_to_construct_nonlocal_conditions
for_higher-order_PE’s_as_well.
Part_III_is_a_comprehensive_introduction_to_modern_numerical_methods_for
time-dependent_elastic_wave_propagation_written_by_P._Joly_and_his_collaborators.
It_consists_of_seven_chapters._In_the_first_one_(Chapter_8)_P._Joly_provides
a_general_introduction_and_an_orientation_to_this_part_of_the_book._In_Chapter
9_he_continues_with_a_presentation_of_the_mathematical_model_for_elastic_wave
propagation,_i.e._the_equations_of_linear_elastodynamics._Chapter_10,_also_by
P._Joly,_is_a_detailed_exposition_of_full_discretizations_of_the_elastodynamics
equations_using_standard_finite_element_subspaces_in_space._Chapter_11,_by_P.
Joly_and_C._Tsogka,_concerns_mixed_finite_element_techniques,_while_Chapter
12,_by_the_same_authors,_covers_fictitious_domain_(finite_element)_methods.
In_Chapter_13,_G._Derveaux,_P._Joly_and_J._Rodr′?guez_analyze_space-time
mesh_refinement_techniques_based_on_the_principle_of_domain_decomposition,
while,_in_Chapter_14,_P._Joly_and_C._Tsogka_review_the_state_of_the_art_of_two
numerical_methods_for_treating_elastic_waves_in_unbounded_media,_namely
3
the_discretization_of_local_absorbing_boundary_conditions_and_the_perfectly
matched_layer_technique.
Part_IV,_by_J._Ekaterinaris,_is_an_overview_of_high-order,_low_numerical_diffusion
numerical_methods_for_complex,_compressible_flows_of_aerodynamics._It
consists_of_five_chapters._Chapter_15_is_introductory_and_Chapter_16_contains
the_governing_equations_of_such_flows._Chapter_17_concerns_high-order_finite
difference_schemes,_Chapter_18_ENO_and_WENO_schemes,_while_Chapter_19
provides_an_introduction_of_Discontinuous_Galerkin_methods_for_hyperbolic
systems.
The_editors_would_like_to_express_their_sincere_thanks_to_the_authors_of_the
various_chapters_for_contributing_their_work_to_this_volume._They_also_wish
to_express_their_sincere_thanks_to_their_students_G._Arabatzis,_I._Toulopoulos,
and_S._Volanis_for_their_help_in_the_preparation_of_this_volume.
Heraklion,_June_2007
N._A._Kampanis
V._A._Dougalis
J._A._Ekaterinaris

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